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Properties

Label 16-3332e8-1.1-c1e8-0-2
Degree 1616
Conductor 1.519×10281.519\times 10^{28}
Sign 11
Analytic cond. 2.51105×10112.51105\times 10^{11}
Root an. cond. 5.158115.15811
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·5-s − 4·11-s + 20·13-s + 16·15-s − 8·17-s + 8·19-s + 4·23-s − 8·25-s − 16·27-s − 16·29-s − 8·31-s − 16·33-s + 8·37-s + 80·39-s + 12·41-s − 4·43-s + 4·47-s − 32·51-s − 16·55-s + 32·57-s − 16·59-s + 32·61-s + 80·65-s + 16·69-s + 24·73-s − 32·75-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s − 1.20·11-s + 5.54·13-s + 4.13·15-s − 1.94·17-s + 1.83·19-s + 0.834·23-s − 8/5·25-s − 3.07·27-s − 2.97·29-s − 1.43·31-s − 2.78·33-s + 1.31·37-s + 12.8·39-s + 1.87·41-s − 0.609·43-s + 0.583·47-s − 4.48·51-s − 2.15·55-s + 4.23·57-s − 2.08·59-s + 4.09·61-s + 9.92·65-s + 1.92·69-s + 2.80·73-s − 3.69·75-s + ⋯

Functional equation

Λ(s)=((216⋅716⋅178)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅716⋅178)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 216⋅716⋅1782^{16} \cdot 7^{16} \cdot 17^{8}
Sign: 11
Analytic conductor: 2.51105×10112.51105\times 10^{11}
Root analytic conductor: 5.158115.15811
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅716⋅178, ( :[1/2]8), 1)(16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 96.8949295996.89492959
L(12)L(\frac12) ≈\approx 96.8949295996.89492959
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
17 (1+T)8 ( 1 + T )^{8}
good3 1−4T+16T2−16pT3+125T4−292T5+616T6−1180T7+2165T8−1180pT9+616p2T10−292p3T11+125p4T12−16p6T13+16p6T14−4p7T15+p8T16 1 - 4 T + 16 T^{2} - 16 p T^{3} + 125 T^{4} - 292 T^{5} + 616 T^{6} - 1180 T^{7} + 2165 T^{8} - 1180 p T^{9} + 616 p^{2} T^{10} - 292 p^{3} T^{11} + 125 p^{4} T^{12} - 16 p^{6} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
5 1−4T+24T2−84T3+61pT4−172pT5+496pT6−5972T7+14293T8−5972pT9+496p3T10−172p4T11+61p5T12−84p5T13+24p6T14−4p7T15+p8T16 1 - 4 T + 24 T^{2} - 84 T^{3} + 61 p T^{4} - 172 p T^{5} + 496 p T^{6} - 5972 T^{7} + 14293 T^{8} - 5972 p T^{9} + 496 p^{3} T^{10} - 172 p^{4} T^{11} + 61 p^{5} T^{12} - 84 p^{5} T^{13} + 24 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
11 1+4T+36T2+156T3+772T4+260pT5+11940T6+39604T7+142038T8+39604pT9+11940p2T10+260p4T11+772p4T12+156p5T13+36p6T14+4p7T15+p8T16 1 + 4 T + 36 T^{2} + 156 T^{3} + 772 T^{4} + 260 p T^{5} + 11940 T^{6} + 39604 T^{7} + 142038 T^{8} + 39604 p T^{9} + 11940 p^{2} T^{10} + 260 p^{4} T^{11} + 772 p^{4} T^{12} + 156 p^{5} T^{13} + 36 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
13 1−20T+242T2−2108T3+14688T4−85220T5+425054T6−1848812T7+7093070T8−1848812pT9+425054p2T10−85220p3T11+14688p4T12−2108p5T13+242p6T14−20p7T15+p8T16 1 - 20 T + 242 T^{2} - 2108 T^{3} + 14688 T^{4} - 85220 T^{5} + 425054 T^{6} - 1848812 T^{7} + 7093070 T^{8} - 1848812 p T^{9} + 425054 p^{2} T^{10} - 85220 p^{3} T^{11} + 14688 p^{4} T^{12} - 2108 p^{5} T^{13} + 242 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16}
19 1−8T+130T2−768T3+7300T4−34608T5+246198T6−51080pT7+5606838T8−51080p2T9+246198p2T10−34608p3T11+7300p4T12−768p5T13+130p6T14−8p7T15+p8T16 1 - 8 T + 130 T^{2} - 768 T^{3} + 7300 T^{4} - 34608 T^{5} + 246198 T^{6} - 51080 p T^{7} + 5606838 T^{8} - 51080 p^{2} T^{9} + 246198 p^{2} T^{10} - 34608 p^{3} T^{11} + 7300 p^{4} T^{12} - 768 p^{5} T^{13} + 130 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
23 1−4T+90T2−540T3+4440T4−29276T5+164982T6−942116T7+4535198T8−942116pT9+164982p2T10−29276p3T11+4440p4T12−540p5T13+90p6T14−4p7T15+p8T16 1 - 4 T + 90 T^{2} - 540 T^{3} + 4440 T^{4} - 29276 T^{5} + 164982 T^{6} - 942116 T^{7} + 4535198 T^{8} - 942116 p T^{9} + 164982 p^{2} T^{10} - 29276 p^{3} T^{11} + 4440 p^{4} T^{12} - 540 p^{5} T^{13} + 90 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
29 1+16T+8pT2+2288T3+20820T4+156880T5+1107792T6+6773040T7+39031606T8+6773040pT9+1107792p2T10+156880p3T11+20820p4T12+2288p5T13+8p7T14+16p7T15+p8T16 1 + 16 T + 8 p T^{2} + 2288 T^{3} + 20820 T^{4} + 156880 T^{5} + 1107792 T^{6} + 6773040 T^{7} + 39031606 T^{8} + 6773040 p T^{9} + 1107792 p^{2} T^{10} + 156880 p^{3} T^{11} + 20820 p^{4} T^{12} + 2288 p^{5} T^{13} + 8 p^{7} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16}
31 1+8T+132T2+1148T3+339pT4+76248T5+557672T6+3391732T7+20312877T8+3391732pT9+557672p2T10+76248p3T11+339p5T12+1148p5T13+132p6T14+8p7T15+p8T16 1 + 8 T + 132 T^{2} + 1148 T^{3} + 339 p T^{4} + 76248 T^{5} + 557672 T^{6} + 3391732 T^{7} + 20312877 T^{8} + 3391732 p T^{9} + 557672 p^{2} T^{10} + 76248 p^{3} T^{11} + 339 p^{5} T^{12} + 1148 p^{5} T^{13} + 132 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16}
37 1−8T+262T2−1752T3+30896T4−174584T5+2158362T6−10209672T7+97900174T8−10209672pT9+2158362p2T10−174584p3T11+30896p4T12−1752p5T13+262p6T14−8p7T15+p8T16 1 - 8 T + 262 T^{2} - 1752 T^{3} + 30896 T^{4} - 174584 T^{5} + 2158362 T^{6} - 10209672 T^{7} + 97900174 T^{8} - 10209672 p T^{9} + 2158362 p^{2} T^{10} - 174584 p^{3} T^{11} + 30896 p^{4} T^{12} - 1752 p^{5} T^{13} + 262 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
41 1−12T+252T2−2556T3+31145T4−259212T5+2352656T6−16138476T7+117350157T8−16138476pT9+2352656p2T10−259212p3T11+31145p4T12−2556p5T13+252p6T14−12p7T15+p8T16 1 - 12 T + 252 T^{2} - 2556 T^{3} + 31145 T^{4} - 259212 T^{5} + 2352656 T^{6} - 16138476 T^{7} + 117350157 T^{8} - 16138476 p T^{9} + 2352656 p^{2} T^{10} - 259212 p^{3} T^{11} + 31145 p^{4} T^{12} - 2556 p^{5} T^{13} + 252 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16}
43 1+4T+134T2−32T3+6463T4−33524T5+283988T6−1881912T7+14761517T8−1881912pT9+283988p2T10−33524p3T11+6463p4T12−32p5T13+134p6T14+4p7T15+p8T16 1 + 4 T + 134 T^{2} - 32 T^{3} + 6463 T^{4} - 33524 T^{5} + 283988 T^{6} - 1881912 T^{7} + 14761517 T^{8} - 1881912 p T^{9} + 283988 p^{2} T^{10} - 33524 p^{3} T^{11} + 6463 p^{4} T^{12} - 32 p^{5} T^{13} + 134 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
47 1−4T+212T2−620T3+19692T4−29500T5+1107892T6−256724T7+51399558T8−256724pT9+1107892p2T10−29500p3T11+19692p4T12−620p5T13+212p6T14−4p7T15+p8T16 1 - 4 T + 212 T^{2} - 620 T^{3} + 19692 T^{4} - 29500 T^{5} + 1107892 T^{6} - 256724 T^{7} + 51399558 T^{8} - 256724 p T^{9} + 1107892 p^{2} T^{10} - 29500 p^{3} T^{11} + 19692 p^{4} T^{12} - 620 p^{5} T^{13} + 212 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
53 1+182T2+396T3+18127T4+76688T5+1202396T6+7128472T7+67495349T8+7128472pT9+1202396p2T10+76688p3T11+18127p4T12+396p5T13+182p6T14+p8T16 1 + 182 T^{2} + 396 T^{3} + 18127 T^{4} + 76688 T^{5} + 1202396 T^{6} + 7128472 T^{7} + 67495349 T^{8} + 7128472 p T^{9} + 1202396 p^{2} T^{10} + 76688 p^{3} T^{11} + 18127 p^{4} T^{12} + 396 p^{5} T^{13} + 182 p^{6} T^{14} + p^{8} T^{16}
59 1+16T+308T2+3024T3+35584T4+259920T5+2411692T6+14703504T7+135177774T8+14703504pT9+2411692p2T10+259920p3T11+35584p4T12+3024p5T13+308p6T14+16p7T15+p8T16 1 + 16 T + 308 T^{2} + 3024 T^{3} + 35584 T^{4} + 259920 T^{5} + 2411692 T^{6} + 14703504 T^{7} + 135177774 T^{8} + 14703504 p T^{9} + 2411692 p^{2} T^{10} + 259920 p^{3} T^{11} + 35584 p^{4} T^{12} + 3024 p^{5} T^{13} + 308 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16}
61 1−32T+780T2−13760T3+202241T4−2496952T5+26836056T6−251837332T7+2095384045T8−251837332pT9+26836056p2T10−2496952p3T11+202241p4T12−13760p5T13+780p6T14−32p7T15+p8T16 1 - 32 T + 780 T^{2} - 13760 T^{3} + 202241 T^{4} - 2496952 T^{5} + 26836056 T^{6} - 251837332 T^{7} + 2095384045 T^{8} - 251837332 p T^{9} + 26836056 p^{2} T^{10} - 2496952 p^{3} T^{11} + 202241 p^{4} T^{12} - 13760 p^{5} T^{13} + 780 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16}
67 1+182T2−380T3+20055T4−58520T5+1944308T6−4410912T7+151707053T8−4410912pT9+1944308p2T10−58520p3T11+20055p4T12−380p5T13+182p6T14+p8T16 1 + 182 T^{2} - 380 T^{3} + 20055 T^{4} - 58520 T^{5} + 1944308 T^{6} - 4410912 T^{7} + 151707053 T^{8} - 4410912 p T^{9} + 1944308 p^{2} T^{10} - 58520 p^{3} T^{11} + 20055 p^{4} T^{12} - 380 p^{5} T^{13} + 182 p^{6} T^{14} + p^{8} T^{16}
71 1+282T2−120T3+39300T4+5592T5+3710766T6+2456208T7+279763670T8+2456208pT9+3710766p2T10+5592p3T11+39300p4T12−120p5T13+282p6T14+p8T16 1 + 282 T^{2} - 120 T^{3} + 39300 T^{4} + 5592 T^{5} + 3710766 T^{6} + 2456208 T^{7} + 279763670 T^{8} + 2456208 p T^{9} + 3710766 p^{2} T^{10} + 5592 p^{3} T^{11} + 39300 p^{4} T^{12} - 120 p^{5} T^{13} + 282 p^{6} T^{14} + p^{8} T^{16}
73 1−24T+532T2−7840T3+111593T4−1287208T5+14379232T6−136783428T7+1257242741T8−136783428pT9+14379232p2T10−1287208p3T11+111593p4T12−7840p5T13+532p6T14−24p7T15+p8T16 1 - 24 T + 532 T^{2} - 7840 T^{3} + 111593 T^{4} - 1287208 T^{5} + 14379232 T^{6} - 136783428 T^{7} + 1257242741 T^{8} - 136783428 p T^{9} + 14379232 p^{2} T^{10} - 1287208 p^{3} T^{11} + 111593 p^{4} T^{12} - 7840 p^{5} T^{13} + 532 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16}
79 1−4T+412T2−980T3+78884T4−128196T5+9972860T6−14036628T7+919676534T8−14036628pT9+9972860p2T10−128196p3T11+78884p4T12−980p5T13+412p6T14−4p7T15+p8T16 1 - 4 T + 412 T^{2} - 980 T^{3} + 78884 T^{4} - 128196 T^{5} + 9972860 T^{6} - 14036628 T^{7} + 919676534 T^{8} - 14036628 p T^{9} + 9972860 p^{2} T^{10} - 128196 p^{3} T^{11} + 78884 p^{4} T^{12} - 980 p^{5} T^{13} + 412 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
83 1−28T+590T2−8124T3+100500T4−988060T5+9343514T6−75809692T7+692605078T8−75809692pT9+9343514p2T10−988060p3T11+100500p4T12−8124p5T13+590p6T14−28p7T15+p8T16 1 - 28 T + 590 T^{2} - 8124 T^{3} + 100500 T^{4} - 988060 T^{5} + 9343514 T^{6} - 75809692 T^{7} + 692605078 T^{8} - 75809692 p T^{9} + 9343514 p^{2} T^{10} - 988060 p^{3} T^{11} + 100500 p^{4} T^{12} - 8124 p^{5} T^{13} + 590 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16}
89 1−20T+628T2−9348T3+170452T4−2031132T5+27607732T6−271275276T7+2973835350T8−271275276pT9+27607732p2T10−2031132p3T11+170452p4T12−9348p5T13+628p6T14−20p7T15+p8T16 1 - 20 T + 628 T^{2} - 9348 T^{3} + 170452 T^{4} - 2031132 T^{5} + 27607732 T^{6} - 271275276 T^{7} + 2973835350 T^{8} - 271275276 p T^{9} + 27607732 p^{2} T^{10} - 2031132 p^{3} T^{11} + 170452 p^{4} T^{12} - 9348 p^{5} T^{13} + 628 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16}
97 1−56T+2040T2−52648T3+1089417T4−18429928T5+264789328T6−3243680020T7+34414745117T8−3243680020pT9+264789328p2T10−18429928p3T11+1089417p4T12−52648p5T13+2040p6T14−56p7T15+p8T16 1 - 56 T + 2040 T^{2} - 52648 T^{3} + 1089417 T^{4} - 18429928 T^{5} + 264789328 T^{6} - 3243680020 T^{7} + 34414745117 T^{8} - 3243680020 p T^{9} + 264789328 p^{2} T^{10} - 18429928 p^{3} T^{11} + 1089417 p^{4} T^{12} - 52648 p^{5} T^{13} + 2040 p^{6} T^{14} - 56 p^{7} T^{15} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.66815107272191255448819501807, −3.36722640436760921509794672440, −3.31074229452828336802700565226, −3.19908917824545086579352607243, −3.10503167744979575370749266539, −2.96466516312116932419056124281, −2.96364176519670467990755345573, −2.77501668657397723823843904395, −2.73951632506840970770739499870, −2.27741904580852919683964362800, −2.22998758619064200829577280762, −2.21431248410130303331426350607, −2.12185972904597207072534096091, −2.05236014544039079387587479111, −1.95318806206666422542220214243, −1.88753429925440673377817815430, −1.86967287483556514081064458890, −1.38817457350297941992360574867, −1.37580546528844004142723143907, −1.17040288994534890482874969269, −0.885259736099810619523120277286, −0.794758986820798593238873719151, −0.61832347012825404223812651152, −0.46589661966613965993927666612, −0.43475734108515463502525547193, 0.43475734108515463502525547193, 0.46589661966613965993927666612, 0.61832347012825404223812651152, 0.794758986820798593238873719151, 0.885259736099810619523120277286, 1.17040288994534890482874969269, 1.37580546528844004142723143907, 1.38817457350297941992360574867, 1.86967287483556514081064458890, 1.88753429925440673377817815430, 1.95318806206666422542220214243, 2.05236014544039079387587479111, 2.12185972904597207072534096091, 2.21431248410130303331426350607, 2.22998758619064200829577280762, 2.27741904580852919683964362800, 2.73951632506840970770739499870, 2.77501668657397723823843904395, 2.96364176519670467990755345573, 2.96466516312116932419056124281, 3.10503167744979575370749266539, 3.19908917824545086579352607243, 3.31074229452828336802700565226, 3.36722640436760921509794672440, 3.66815107272191255448819501807

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.