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Properties

Label 12-91e12-1.1-c1e6-0-6
Degree $12$
Conductor $3.225\times 10^{23}$
Sign $1$
Analytic cond. $8.35909\times 10^{10}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·4-s − 6·5-s − 4·8-s − 7·9-s − 24·10-s + 4·11-s − 8·16-s + 4·17-s − 28·18-s − 2·19-s − 24·20-s + 16·22-s − 12·23-s + 8·25-s − 2·27-s − 8·29-s + 14·31-s + 16·34-s − 28·36-s + 12·37-s − 8·38-s + 24·40-s − 28·41-s + 2·43-s + 16·44-s + 42·45-s + ⋯
L(s)  = 1  + 2.82·2-s + 2·4-s − 2.68·5-s − 1.41·8-s − 7/3·9-s − 7.58·10-s + 1.20·11-s − 2·16-s + 0.970·17-s − 6.59·18-s − 0.458·19-s − 5.36·20-s + 3.41·22-s − 2.50·23-s + 8/5·25-s − 0.384·27-s − 1.48·29-s + 2.51·31-s + 2.74·34-s − 4.66·36-s + 1.97·37-s − 1.29·38-s + 3.79·40-s − 4.37·41-s + 0.304·43-s + 2.41·44-s + 6.26·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(8.35909\times 10^{10}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 - p^{2} T + 3 p^{2} T^{2} - 7 p^{2} T^{3} + 7 p^{3} T^{4} - 3 p^{5} T^{5} + 145 T^{6} - 3 p^{6} T^{7} + 7 p^{5} T^{8} - 7 p^{5} T^{9} + 3 p^{6} T^{10} - p^{7} T^{11} + p^{6} T^{12} \)
3 \( 1 + 7 T^{2} + 2 T^{3} + 28 T^{4} - 2 T^{5} + 100 T^{6} - 2 p T^{7} + 28 p^{2} T^{8} + 2 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + 6 T + 28 T^{2} + 4 p^{2} T^{3} + 333 T^{4} + 878 T^{5} + 2121 T^{6} + 878 p T^{7} + 333 p^{2} T^{8} + 4 p^{5} T^{9} + 28 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T + 49 T^{2} - 204 T^{3} + 1152 T^{4} - 4240 T^{5} + 16164 T^{6} - 4240 p T^{7} + 1152 p^{2} T^{8} - 204 p^{3} T^{9} + 49 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T + 81 T^{2} - 280 T^{3} + 3074 T^{4} - 8724 T^{5} + 67033 T^{6} - 8724 p T^{7} + 3074 p^{2} T^{8} - 280 p^{3} T^{9} + 81 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T + 87 T^{2} + 10 p T^{3} + 3512 T^{4} + 7188 T^{5} + 84124 T^{6} + 7188 p T^{7} + 3512 p^{2} T^{8} + 10 p^{4} T^{9} + 87 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 118 T^{2} + 772 T^{3} + 4479 T^{4} + 1000 p T^{5} + 111732 T^{6} + 1000 p^{2} T^{7} + 4479 p^{2} T^{8} + 772 p^{3} T^{9} + 118 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 8 T + 130 T^{2} + 594 T^{3} + 6099 T^{4} + 18374 T^{5} + 187029 T^{6} + 18374 p T^{7} + 6099 p^{2} T^{8} + 594 p^{3} T^{9} + 130 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 14 T + 216 T^{2} - 1972 T^{3} + 17840 T^{4} - 117486 T^{5} + 749554 T^{6} - 117486 p T^{7} + 17840 p^{2} T^{8} - 1972 p^{3} T^{9} + 216 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 12 T + 135 T^{2} - 1176 T^{3} + 10575 T^{4} - 70860 T^{5} + 472322 T^{6} - 70860 p T^{7} + 10575 p^{2} T^{8} - 1176 p^{3} T^{9} + 135 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 28 T + 503 T^{2} + 6376 T^{3} + 64135 T^{4} + 528684 T^{5} + 3676082 T^{6} + 528684 p T^{7} + 64135 p^{2} T^{8} + 6376 p^{3} T^{9} + 503 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 2 T + 149 T^{2} - 340 T^{3} + 11408 T^{4} - 21594 T^{5} + 590652 T^{6} - 21594 p T^{7} + 11408 p^{2} T^{8} - 340 p^{3} T^{9} + 149 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 244 T^{2} + 2532 T^{3} + 25548 T^{4} + 209078 T^{5} + 1534242 T^{6} + 209078 p T^{7} + 25548 p^{2} T^{8} + 2532 p^{3} T^{9} + 244 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 22 T + 409 T^{2} + 5130 T^{3} + 58074 T^{4} + 513982 T^{5} + 4153497 T^{6} + 513982 p T^{7} + 58074 p^{2} T^{8} + 5130 p^{3} T^{9} + 409 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 192 T^{2} - 380 T^{3} + 20828 T^{4} - 39642 T^{5} + 1464142 T^{6} - 39642 p T^{7} + 20828 p^{2} T^{8} - 380 p^{3} T^{9} + 192 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 14 T + 279 T^{2} - 2854 T^{3} + 32699 T^{4} - 263412 T^{5} + 2369290 T^{6} - 263412 p T^{7} + 32699 p^{2} T^{8} - 2854 p^{3} T^{9} + 279 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 24 T + 496 T^{2} - 6632 T^{3} + 80672 T^{4} - 775096 T^{5} + 6983258 T^{6} - 775096 p T^{7} + 80672 p^{2} T^{8} - 6632 p^{3} T^{9} + 496 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 4 T + 358 T^{2} - 1164 T^{3} + 57375 T^{4} - 149608 T^{5} + 5246868 T^{6} - 149608 p T^{7} + 57375 p^{2} T^{8} - 1164 p^{3} T^{9} + 358 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 36 T + 919 T^{2} + 15928 T^{3} + 225311 T^{4} + 2516900 T^{5} + 23841506 T^{6} + 2516900 p T^{7} + 225311 p^{2} T^{8} + 15928 p^{3} T^{9} + 919 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 28 T + 686 T^{2} + 11252 T^{3} + 159023 T^{4} + 1794648 T^{5} + 17548548 T^{6} + 1794648 p T^{7} + 159023 p^{2} T^{8} + 11252 p^{3} T^{9} + 686 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 26 T + 684 T^{2} + 10460 T^{3} + 156752 T^{4} + 1683426 T^{5} + 17728366 T^{6} + 1683426 p T^{7} + 156752 p^{2} T^{8} + 10460 p^{3} T^{9} + 684 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 42 T + 1087 T^{2} + 20330 T^{3} + 304908 T^{4} + 3736348 T^{5} + 38506140 T^{6} + 3736348 p T^{7} + 304908 p^{2} T^{8} + 20330 p^{3} T^{9} + 1087 p^{4} T^{10} + 42 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 24 T + 7 p T^{2} + 10706 T^{3} + 174956 T^{4} + 2000590 T^{5} + 22996700 T^{6} + 2000590 p T^{7} + 174956 p^{2} T^{8} + 10706 p^{3} T^{9} + 7 p^{5} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.30268661213362869590398978194, −4.06665339416031601646052520914, −3.97328844641567166160062141247, −3.96846209759311848304652081573, −3.90858237544157449070383295178, −3.89529424317487692721605175534, −3.86204610130443931281105927961, −3.60128353699397280344739453950, −3.40889028435890523361996866388, −3.19663761387544273683961347761, −3.11954435105357791422353975340, −3.09876182172350110081049054116, −2.83651920245197985286762686248, −2.83593047008111530465754053218, −2.69270970697352172101059046908, −2.47212180377614272715276692785, −2.33648980158994730450071408062, −2.12969152065884468852853984017, −1.70763558118708614109791082865, −1.64794104436450507519480604084, −1.59664737271179248295374662117, −1.44092533288954876886338514966, −1.23154249270172932365943367700, −1.03958880402848548919942717092, −0.808837688041672869830498209875, 0, 0, 0, 0, 0, 0, 0.808837688041672869830498209875, 1.03958880402848548919942717092, 1.23154249270172932365943367700, 1.44092533288954876886338514966, 1.59664737271179248295374662117, 1.64794104436450507519480604084, 1.70763558118708614109791082865, 2.12969152065884468852853984017, 2.33648980158994730450071408062, 2.47212180377614272715276692785, 2.69270970697352172101059046908, 2.83593047008111530465754053218, 2.83651920245197985286762686248, 3.09876182172350110081049054116, 3.11954435105357791422353975340, 3.19663761387544273683961347761, 3.40889028435890523361996866388, 3.60128353699397280344739453950, 3.86204610130443931281105927961, 3.89529424317487692721605175534, 3.90858237544157449070383295178, 3.96846209759311848304652081573, 3.97328844641567166160062141247, 4.06665339416031601646052520914, 4.30268661213362869590398978194

Graph of the $Z$-function along the critical line

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