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Properties

Label 32-637e16-1.1-c1e16-0-4
Degree 3232
Conductor 7.349×10447.349\times 10^{44}
Sign 11
Analytic cond. 2.00754×10112.00754\times 10^{11}
Root an. cond. 2.255322.25532
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  + 4·2-s + 10·4-s + 32·8-s + 10·9-s − 4·11-s + 87·16-s + 40·18-s − 16·22-s + 12·23-s − 52·25-s + 8·29-s + 196·32-s + 100·36-s − 8·37-s + 32·43-s − 40·44-s + 48·46-s − 208·50-s − 8·53-s + 32·58-s + 438·64-s + 20·67-s + 8·71-s + 320·72-s − 32·74-s − 8·79-s + 67·81-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 11.3·8-s + 10/3·9-s − 1.20·11-s + 87/4·16-s + 9.42·18-s − 3.41·22-s + 2.50·23-s − 10.3·25-s + 1.48·29-s + 34.6·32-s + 50/3·36-s − 1.31·37-s + 4.87·43-s − 6.03·44-s + 7.07·46-s − 29.4·50-s − 1.09·53-s + 4.20·58-s + 54.7·64-s + 2.44·67-s + 0.949·71-s + 37.7·72-s − 3.71·74-s − 0.900·79-s + 67/9·81-s + ⋯

Functional equation

Λ(s)=((732⋅1316)s/2 ΓC(s)16 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((732⋅1316)s/2 ΓC(s+1/2)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 732⋅13167^{32} \cdot 13^{16}
Sign: 11
Analytic conductor: 2.00754×10112.00754\times 10^{11}
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 732⋅1316, ( :[1/2]16), 1)(32,\ 7^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 694.6913152694.6913152
L(12)L(\frac12) ≈\approx 694.6913152694.6913152
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)
bad7 1 1
13 1+34T2+430T4+4984T6+76567T8+4984p2T10+430p4T12+34p6T14+p8T16 1 + 34 T^{2} + 430 T^{4} + 4984 T^{6} + 76567 T^{8} + 4984 p^{2} T^{10} + 430 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16}
good2 (1−pT+T2−3pT3+5pT4−p2T5+27T6−5p3T7+13T8−5p4T9+27p2T10−p5T11+5p5T12−3p6T13+p6T14−p8T15+p8T16)2 ( 1 - p T + T^{2} - 3 p T^{3} + 5 p T^{4} - p^{2} T^{5} + 27 T^{6} - 5 p^{3} T^{7} + 13 T^{8} - 5 p^{4} T^{9} + 27 p^{2} T^{10} - p^{5} T^{11} + 5 p^{5} T^{12} - 3 p^{6} T^{13} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2}
3 1−10T2+11pT4−58T6+124pT8−2116T10+6131T12−1774p2T14+50155T16−1774p4T18+6131p4T20−2116p6T22+124p9T24−58p10T26+11p13T28−10p14T30+p16T32 1 - 10 T^{2} + 11 p T^{4} - 58 T^{6} + 124 p T^{8} - 2116 T^{10} + 6131 T^{12} - 1774 p^{2} T^{14} + 50155 T^{16} - 1774 p^{4} T^{18} + 6131 p^{4} T^{20} - 2116 p^{6} T^{22} + 124 p^{9} T^{24} - 58 p^{10} T^{26} + 11 p^{13} T^{28} - 10 p^{14} T^{30} + p^{16} T^{32}
5 (1+26T2+311T4+2358T6+13281T8+2358p2T10+311p4T12+26p6T14+p8T16)2 ( 1 + 26 T^{2} + 311 T^{4} + 2358 T^{6} + 13281 T^{8} + 2358 p^{2} T^{10} + 311 p^{4} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16} )^{2}
11 (1+2T−35T2−30T3+802T4+256T5−12663T6−1238T7+153211T8−1238pT9−12663p2T10+256p3T11+802p4T12−30p5T13−35p6T14+2p7T15+p8T16)2 ( 1 + 2 T - 35 T^{2} - 30 T^{3} + 802 T^{4} + 256 T^{5} - 12663 T^{6} - 1238 T^{7} + 153211 T^{8} - 1238 p T^{9} - 12663 p^{2} T^{10} + 256 p^{3} T^{11} + 802 p^{4} T^{12} - 30 p^{5} T^{13} - 35 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2}
17 1−78T2+2942T4−74036T6+1506893T8−29044204T10+585942562T12−11972588178T14+220591800628T16−11972588178p2T18+585942562p4T20−29044204p6T22+1506893p8T24−74036p10T26+2942p12T28−78p14T30+p16T32 1 - 78 T^{2} + 2942 T^{4} - 74036 T^{6} + 1506893 T^{8} - 29044204 T^{10} + 585942562 T^{12} - 11972588178 T^{14} + 220591800628 T^{16} - 11972588178 p^{2} T^{18} + 585942562 p^{4} T^{20} - 29044204 p^{6} T^{22} + 1506893 p^{8} T^{24} - 74036 p^{10} T^{26} + 2942 p^{12} T^{28} - 78 p^{14} T^{30} + p^{16} T^{32}
19 1−58T2+1661T4−9818T6−676340T8+23829108T10−206707529T12−5249133354T14+201079481139T16−5249133354p2T18−206707529p4T20+23829108p6T22−676340p8T24−9818p10T26+1661p12T28−58p14T30+p16T32 1 - 58 T^{2} + 1661 T^{4} - 9818 T^{6} - 676340 T^{8} + 23829108 T^{10} - 206707529 T^{12} - 5249133354 T^{14} + 201079481139 T^{16} - 5249133354 p^{2} T^{18} - 206707529 p^{4} T^{20} + 23829108 p^{6} T^{22} - 676340 p^{8} T^{24} - 9818 p^{10} T^{26} + 1661 p^{12} T^{28} - 58 p^{14} T^{30} + p^{16} T^{32}
23 (1−6T−42T2+248T3+1306T4−4598T5−43856T6+22390T7+1346875T8+22390pT9−43856p2T10−4598p3T11+1306p4T12+248p5T13−42p6T14−6p7T15+p8T16)2 ( 1 - 6 T - 42 T^{2} + 248 T^{3} + 1306 T^{4} - 4598 T^{5} - 43856 T^{6} + 22390 T^{7} + 1346875 T^{8} + 22390 p T^{9} - 43856 p^{2} T^{10} - 4598 p^{3} T^{11} + 1306 p^{4} T^{12} + 248 p^{5} T^{13} - 42 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2}
29 (1−4T−15T2+192T3−1568T4+5696T5−pT6−217482T7+2139619T8−217482pT9−p3T10+5696p3T11−1568p4T12+192p5T13−15p6T14−4p7T15+p8T16)2 ( 1 - 4 T - 15 T^{2} + 192 T^{3} - 1568 T^{4} + 5696 T^{5} - p T^{6} - 217482 T^{7} + 2139619 T^{8} - 217482 p T^{9} - p^{3} T^{10} + 5696 p^{3} T^{11} - 1568 p^{4} T^{12} + 192 p^{5} T^{13} - 15 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2}
31 (1+178T2+15406T4+835752T6+30991415T8+835752p2T10+15406p4T12+178p6T14+p8T16)2 ( 1 + 178 T^{2} + 15406 T^{4} + 835752 T^{6} + 30991415 T^{8} + 835752 p^{2} T^{10} + 15406 p^{4} T^{12} + 178 p^{6} T^{14} + p^{8} T^{16} )^{2}
37 (1+4T−46T2+152T3+2182T4−7008T5+73816T6+392460T7−3440049T8+392460pT9+73816p2T10−7008p3T11+2182p4T12+152p5T13−46p6T14+4p7T15+p8T16)2 ( 1 + 4 T - 46 T^{2} + 152 T^{3} + 2182 T^{4} - 7008 T^{5} + 73816 T^{6} + 392460 T^{7} - 3440049 T^{8} + 392460 p T^{9} + 73816 p^{2} T^{10} - 7008 p^{3} T^{11} + 2182 p^{4} T^{12} + 152 p^{5} T^{13} - 46 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2}
41 1−152T2+16324T4−1214800T6+75014282T8−3793839848T10+172337071760T12−7119768845032T14+293658219524371T16−7119768845032p2T18+172337071760p4T20−3793839848p6T22+75014282p8T24−1214800p10T26+16324p12T28−152p14T30+p16T32 1 - 152 T^{2} + 16324 T^{4} - 1214800 T^{6} + 75014282 T^{8} - 3793839848 T^{10} + 172337071760 T^{12} - 7119768845032 T^{14} + 293658219524371 T^{16} - 7119768845032 p^{2} T^{18} + 172337071760 p^{4} T^{20} - 3793839848 p^{6} T^{22} + 75014282 p^{8} T^{24} - 1214800 p^{10} T^{26} + 16324 p^{12} T^{28} - 152 p^{14} T^{30} + p^{16} T^{32}
43 (1−16T+99T2−472T3+1730T4+7168T5−116817T6+1124260T7−9477617T8+1124260pT9−116817p2T10+7168p3T11+1730p4T12−472p5T13+99p6T14−16p7T15+p8T16)2 ( 1 - 16 T + 99 T^{2} - 472 T^{3} + 1730 T^{4} + 7168 T^{5} - 116817 T^{6} + 1124260 T^{7} - 9477617 T^{8} + 1124260 p T^{9} - 116817 p^{2} T^{10} + 7168 p^{3} T^{11} + 1730 p^{4} T^{12} - 472 p^{5} T^{13} + 99 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2}
47 (1+150T2+11494T4+707936T6+37180919T8+707936p2T10+11494p4T12+150p6T14+p8T16)2 ( 1 + 150 T^{2} + 11494 T^{4} + 707936 T^{6} + 37180919 T^{8} + 707936 p^{2} T^{10} + 11494 p^{4} T^{12} + 150 p^{6} T^{14} + p^{8} T^{16} )^{2}
53 (1+2T+132T2+248T3+8645T4+248pT5+132p2T6+2p3T7+p4T8)4 ( 1 + 2 T + 132 T^{2} + 248 T^{3} + 8645 T^{4} + 248 p T^{5} + 132 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4}
59 1−278T2+40998T4−3814004T6+237186965T8−9386021884T10+154854604410T12+7674718497878T14−788274714787388T16+7674718497878p2T18+154854604410p4T20−9386021884p6T22+237186965p8T24−3814004p10T26+40998p12T28−278p14T30+p16T32 1 - 278 T^{2} + 40998 T^{4} - 3814004 T^{6} + 237186965 T^{8} - 9386021884 T^{10} + 154854604410 T^{12} + 7674718497878 T^{14} - 788274714787388 T^{16} + 7674718497878 p^{2} T^{18} + 154854604410 p^{4} T^{20} - 9386021884 p^{6} T^{22} + 237186965 p^{8} T^{24} - 3814004 p^{10} T^{26} + 40998 p^{12} T^{28} - 278 p^{14} T^{30} + p^{16} T^{32}
61 1−380T2+76544T4−10897784T6+1227826114T8−116331500276T10+9578956327264T12−696692904086100T14+45045833784892083T16−696692904086100p2T18+9578956327264p4T20−116331500276p6T22+1227826114p8T24−10897784p10T26+76544p12T28−380p14T30+p16T32 1 - 380 T^{2} + 76544 T^{4} - 10897784 T^{6} + 1227826114 T^{8} - 116331500276 T^{10} + 9578956327264 T^{12} - 696692904086100 T^{14} + 45045833784892083 T^{16} - 696692904086100 p^{2} T^{18} + 9578956327264 p^{4} T^{20} - 116331500276 p^{6} T^{22} + 1227826114 p^{8} T^{24} - 10897784 p^{10} T^{26} + 76544 p^{12} T^{28} - 380 p^{14} T^{30} + p^{16} T^{32}
67 (1−10T−162T2+956T3+24917T4−71312T5−2420694T6+1789858T7+185730724T8+1789858pT9−2420694p2T10−71312p3T11+24917p4T12+956p5T13−162p6T14−10p7T15+p8T16)2 ( 1 - 10 T - 162 T^{2} + 956 T^{3} + 24917 T^{4} - 71312 T^{5} - 2420694 T^{6} + 1789858 T^{7} + 185730724 T^{8} + 1789858 p T^{9} - 2420694 p^{2} T^{10} - 71312 p^{3} T^{11} + 24917 p^{4} T^{12} + 956 p^{5} T^{13} - 162 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2}
71 (1−4T−102T2+696T3−1042T4−9336T5−125464T6−1401412T7+51152511T8−1401412pT9−125464p2T10−9336p3T11−1042p4T12+696p5T13−102p6T14−4p7T15+p8T16)2 ( 1 - 4 T - 102 T^{2} + 696 T^{3} - 1042 T^{4} - 9336 T^{5} - 125464 T^{6} - 1401412 T^{7} + 51152511 T^{8} - 1401412 p T^{9} - 125464 p^{2} T^{10} - 9336 p^{3} T^{11} - 1042 p^{4} T^{12} + 696 p^{5} T^{13} - 102 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2}
73 (1+156T2+7232T4+2020pT6+11253390T8+2020p3T10+7232p4T12+156p6T14+p8T16)2 ( 1 + 156 T^{2} + 7232 T^{4} + 2020 p T^{6} + 11253390 T^{8} + 2020 p^{3} T^{10} + 7232 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} )^{2}
79 (1+2T+38T2−238T3+1686T4−238pT5+38p2T6+2p3T7+p4T8)4 ( 1 + 2 T + 38 T^{2} - 238 T^{3} + 1686 T^{4} - 238 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4}
83 (1+314T2+46430T4+4837464T6+427877847T8+4837464p2T10+46430p4T12+314p6T14+p8T16)2 ( 1 + 314 T^{2} + 46430 T^{4} + 4837464 T^{6} + 427877847 T^{8} + 4837464 p^{2} T^{10} + 46430 p^{4} T^{12} + 314 p^{6} T^{14} + p^{8} T^{16} )^{2}
89 1−106T2−6635T4+2260150T6−56645140T8−20540045260T10+2028500040815T12+92860376641870T14−22701882367538285T16+92860376641870p2T18+2028500040815p4T20−20540045260p6T22−56645140p8T24+2260150p10T26−6635p12T28−106p14T30+p16T32 1 - 106 T^{2} - 6635 T^{4} + 2260150 T^{6} - 56645140 T^{8} - 20540045260 T^{10} + 2028500040815 T^{12} + 92860376641870 T^{14} - 22701882367538285 T^{16} + 92860376641870 p^{2} T^{18} + 2028500040815 p^{4} T^{20} - 20540045260 p^{6} T^{22} - 56645140 p^{8} T^{24} + 2260150 p^{10} T^{26} - 6635 p^{12} T^{28} - 106 p^{14} T^{30} + p^{16} T^{32}
97 1−412T2+87509T4−11349760T6+907888036T8−32651756224T10−1974791695085T12+422191622326362T14−45739383615060693T16+422191622326362p2T18−1974791695085p4T20−32651756224p6T22+907888036p8T24−11349760p10T26+87509p12T28−412p14T30+p16T32 1 - 412 T^{2} + 87509 T^{4} - 11349760 T^{6} + 907888036 T^{8} - 32651756224 T^{10} - 1974791695085 T^{12} + 422191622326362 T^{14} - 45739383615060693 T^{16} + 422191622326362 p^{2} T^{18} - 1974791695085 p^{4} T^{20} - 32651756224 p^{6} T^{22} + 907888036 p^{8} T^{24} - 11349760 p^{10} T^{26} + 87509 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32}
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−2.82971545980281424348696263257, −2.71465563260493320135431246918, −2.53676864066042077082901779177, −2.50572230772502125393385968961, −2.50453516339212674287377002269, −2.40507548721078497229187371039, −2.35079628947152365897669539918, −2.34584168565053353127241613396, −2.11912395278826651169602815346, −2.01061848449569760489930310482, −1.94674195769371218729679271588, −1.87272041631664334366659867596, −1.80515769542522829470395819796, −1.66158221927596276142838311504, −1.59058216955735166432475461521, −1.47572498512007748167774136262, −1.46984545366465835967030062283, −1.42568431320038204824636494391, −1.40864519121509738458608423867, −1.39684944977077239307035708651, −0.71053457165757075059635209742, −0.68533658273075388308636848416, −0.68030151007276984111322096984, −0.57490128015347619171821733892, −0.57451372504531519107677728228, 0.57451372504531519107677728228, 0.57490128015347619171821733892, 0.68030151007276984111322096984, 0.68533658273075388308636848416, 0.71053457165757075059635209742, 1.39684944977077239307035708651, 1.40864519121509738458608423867, 1.42568431320038204824636494391, 1.46984545366465835967030062283, 1.47572498512007748167774136262, 1.59058216955735166432475461521, 1.66158221927596276142838311504, 1.80515769542522829470395819796, 1.87272041631664334366659867596, 1.94674195769371218729679271588, 2.01061848449569760489930310482, 2.11912395278826651169602815346, 2.34584168565053353127241613396, 2.35079628947152365897669539918, 2.40507548721078497229187371039, 2.50453516339212674287377002269, 2.50572230772502125393385968961, 2.53676864066042077082901779177, 2.71465563260493320135431246918, 2.82971545980281424348696263257

Graph of the ZZ-function along the critical line

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