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Properties

Label 12-1323e6-1.1-c3e6-0-4
Degree 1212
Conductor 5.362×10185.362\times 10^{18}
Sign 11
Analytic cond. 2.26232×10112.26232\times 10^{11}
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 66

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 52·13-s − 69·16-s + 62·19-s − 398·25-s + 82·31-s − 1.13e3·37-s − 1.56e3·43-s + 416·52-s − 886·61-s + 766·64-s − 2.08e3·67-s + 2.39e3·73-s − 496·76-s + 984·79-s + 682·97-s + 3.18e3·100-s + 3.02e3·103-s − 2.47e3·109-s − 1.94e3·121-s − 656·124-s + 127-s + 131-s + 137-s + 139-s + 9.05e3·148-s + 149-s + ⋯
L(s)  = 1  − 4-s − 1.10·13-s − 1.07·16-s + 0.748·19-s − 3.18·25-s + 0.475·31-s − 5.02·37-s − 5.55·43-s + 1.10·52-s − 1.85·61-s + 1.49·64-s − 3.80·67-s + 3.84·73-s − 0.748·76-s + 1.40·79-s + 0.713·97-s + 3.18·100-s + 2.89·103-s − 2.17·109-s − 1.46·121-s − 0.475·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.02·148-s + 0.000549·149-s + ⋯

Functional equation

Λ(s)=((318⋅712)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((318⋅712)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 318⋅7123^{18} \cdot 7^{12}
Sign: 11
Analytic conductor: 2.26232×10112.26232\times 10^{11}
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 318⋅712, ( :[3/2]6), 1)(12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
7 1 1
good2 1+p3T2+133T4+425pT6+133p6T8+p15T10+p18T12 1 + p^{3} T^{2} + 133 T^{4} + 425 p T^{6} + 133 p^{6} T^{8} + p^{15} T^{10} + p^{18} T^{12}
5 1+398T2+88267T4+13515052T6+88267p6T8+398p12T10+p18T12 1 + 398 T^{2} + 88267 T^{4} + 13515052 T^{6} + 88267 p^{6} T^{8} + 398 p^{12} T^{10} + p^{18} T^{12}
11 1+1946T2+1291819T4−782053340T6+1291819p6T8+1946p12T10+p18T12 1 + 1946 T^{2} + 1291819 T^{4} - 782053340 T^{6} + 1291819 p^{6} T^{8} + 1946 p^{12} T^{10} + p^{18} T^{12}
13 (1+2pT+20p2T2+150062T3+20p5T4+2p7T5+p9T6)2 ( 1 + 2 p T + 20 p^{2} T^{2} + 150062 T^{3} + 20 p^{5} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} )^{2}
17 1+19638T2+193457043T4+1185167951260T6+193457043p6T8+19638p12T10+p18T12 1 + 19638 T^{2} + 193457043 T^{4} + 1185167951260 T^{6} + 193457043 p^{6} T^{8} + 19638 p^{12} T^{10} + p^{18} T^{12}
19 (1−31T+11177T2−354214T3+11177p3T4−31p6T5+p9T6)2 ( 1 - 31 T + 11177 T^{2} - 354214 T^{3} + 11177 p^{3} T^{4} - 31 p^{6} T^{5} + p^{9} T^{6} )^{2}
23 1+26874T2+137157615T4−1070796212084T6+137157615p6T8+26874p12T10+p18T12 1 + 26874 T^{2} + 137157615 T^{4} - 1070796212084 T^{6} + 137157615 p^{6} T^{8} + 26874 p^{12} T^{10} + p^{18} T^{12}
29 1+107486T2+5558833147T4+171068655050380T6+5558833147p6T8+107486p12T10+p18T12 1 + 107486 T^{2} + 5558833147 T^{4} + 171068655050380 T^{6} + 5558833147 p^{6} T^{8} + 107486 p^{12} T^{10} + p^{18} T^{12}
31 (1−41T+55666T2−237883T3+55666p3T4−41p6T5+p9T6)2 ( 1 - 41 T + 55666 T^{2} - 237883 T^{3} + 55666 p^{3} T^{4} - 41 p^{6} T^{5} + p^{9} T^{6} )^{2}
37 (1+566T+246688T2+61584934T3+246688p3T4+566p6T5+p9T6)2 ( 1 + 566 T + 246688 T^{2} + 61584934 T^{3} + 246688 p^{3} T^{4} + 566 p^{6} T^{5} + p^{9} T^{6} )^{2}
41 1+201446T2+24090872899T4+1992163622083516T6+24090872899p6T8+201446p12T10+p18T12 1 + 201446 T^{2} + 24090872899 T^{4} + 1992163622083516 T^{6} + 24090872899 p^{6} T^{8} + 201446 p^{12} T^{10} + p^{18} T^{12}
43 (1+783T+433368T2+140027659T3+433368p3T4+783p6T5+p9T6)2 ( 1 + 783 T + 433368 T^{2} + 140027659 T^{3} + 433368 p^{3} T^{4} + 783 p^{6} T^{5} + p^{9} T^{6} )^{2}
47 1+127818T2+18065263059T4+2679796621113892T6+18065263059p6T8+127818p12T10+p18T12 1 + 127818 T^{2} + 18065263059 T^{4} + 2679796621113892 T^{6} + 18065263059 p^{6} T^{8} + 127818 p^{12} T^{10} + p^{18} T^{12}
53 1+680526T2+219567635127T4+41561598578306884T6+219567635127p6T8+680526p12T10+p18T12 1 + 680526 T^{2} + 219567635127 T^{4} + 41561598578306884 T^{6} + 219567635127 p^{6} T^{8} + 680526 p^{12} T^{10} + p^{18} T^{12}
59 1+252810T2+118459786923T4+17779226383468420T6+118459786923p6T8+252810p12T10+p18T12 1 + 252810 T^{2} + 118459786923 T^{4} + 17779226383468420 T^{6} + 118459786923 p^{6} T^{8} + 252810 p^{12} T^{10} + p^{18} T^{12}
61 (1+443T+203324T2−18833347T3+203324p3T4+443p6T5+p9T6)2 ( 1 + 443 T + 203324 T^{2} - 18833347 T^{3} + 203324 p^{3} T^{4} + 443 p^{6} T^{5} + p^{9} T^{6} )^{2}
67 (1+1042T+867758T2+445253716T3+867758p3T4+1042p6T5+p9T6)2 ( 1 + 1042 T + 867758 T^{2} + 445253716 T^{3} + 867758 p^{3} T^{4} + 1042 p^{6} T^{5} + p^{9} T^{6} )^{2}
71 1+1054578T2+491902090755T4+2420618974688108pT6+491902090755p6T8+1054578p12T10+p18T12 1 + 1054578 T^{2} + 491902090755 T^{4} + 2420618974688108 p T^{6} + 491902090755 p^{6} T^{8} + 1054578 p^{12} T^{10} + p^{18} T^{12}
73 (1−1199T+1316443T2−794269678T3+1316443p3T4−1199p6T5+p9T6)2 ( 1 - 1199 T + 1316443 T^{2} - 794269678 T^{3} + 1316443 p^{3} T^{4} - 1199 p^{6} T^{5} + p^{9} T^{6} )^{2}
79 (1−492T+608946T2−546599468T3+608946p3T4−492p6T5+p9T6)2 ( 1 - 492 T + 608946 T^{2} - 546599468 T^{3} + 608946 p^{3} T^{4} - 492 p^{6} T^{5} + p^{9} T^{6} )^{2}
83 1−834486T2+1207789948167T4−565854790583996660T6+1207789948167p6T8−834486p12T10+p18T12 1 - 834486 T^{2} + 1207789948167 T^{4} - 565854790583996660 T^{6} + 1207789948167 p^{6} T^{8} - 834486 p^{12} T^{10} + p^{18} T^{12}
89 1+949814T2+691701373123T4+216133792389242716T6+691701373123p6T8+949814p12T10+p18T12 1 + 949814 T^{2} + 691701373123 T^{4} + 216133792389242716 T^{6} + 691701373123 p^{6} T^{8} + 949814 p^{12} T^{10} + p^{18} T^{12}
97 (1−341T+490090T2+960147395T3+490090p3T4−341p6T5+p9T6)2 ( 1 - 341 T + 490090 T^{2} + 960147395 T^{3} + 490090 p^{3} T^{4} - 341 p^{6} T^{5} + p^{9} T^{6} )^{2}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.27290667958710477924052552893, −4.88359239990992956705299543976, −4.72868000510255781844456117816, −4.66973403249598545827425700627, −4.63604177812513844000713721965, −4.42080721764398605479863312735, −4.41750350100876189168198675584, −3.78788673680624213131392684620, −3.71854616458668008537437990115, −3.67467351083477055177122819194, −3.50180763645388964696767481365, −3.44453225813179184607177230420, −3.44245629685745685875045094732, −3.13932692664993126265403130134, −2.82807773068587826098911251768, −2.59066875326978552986862920174, −2.33702165415097516486525467925, −2.22537825151505059931539706123, −1.94444790600979433188081345551, −1.92201750575789416896751015963, −1.91419219511509351460931145021, −1.31880122532995929599016682212, −1.28165342310341055228676965163, −1.20937597270489172291112748589, −0.910064134066559161033882164644, 0, 0, 0, 0, 0, 0, 0.910064134066559161033882164644, 1.20937597270489172291112748589, 1.28165342310341055228676965163, 1.31880122532995929599016682212, 1.91419219511509351460931145021, 1.92201750575789416896751015963, 1.94444790600979433188081345551, 2.22537825151505059931539706123, 2.33702165415097516486525467925, 2.59066875326978552986862920174, 2.82807773068587826098911251768, 3.13932692664993126265403130134, 3.44245629685745685875045094732, 3.44453225813179184607177230420, 3.50180763645388964696767481365, 3.67467351083477055177122819194, 3.71854616458668008537437990115, 3.78788673680624213131392684620, 4.41750350100876189168198675584, 4.42080721764398605479863312735, 4.63604177812513844000713721965, 4.66973403249598545827425700627, 4.72868000510255781844456117816, 4.88359239990992956705299543976, 5.27290667958710477924052552893

Graph of the ZZ-function along the critical line

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