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Properties

Label 12-637e6-1.1-c1e6-0-2
Degree $12$
Conductor $6.681\times 10^{16}$
Sign $1$
Analytic cond. $17318.0$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 2·4-s + 2·5-s + 2·6-s − 8-s + 3·9-s − 2·10-s − 2·11-s − 4·12-s − 6·13-s − 4·15-s + 3·16-s + 4·17-s − 3·18-s − 4·19-s + 4·20-s + 2·22-s − 10·23-s + 2·24-s + 12·25-s + 6·26-s − 2·27-s + 48·29-s + 4·30-s − 4·31-s − 2·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 4-s + 0.894·5-s + 0.816·6-s − 0.353·8-s + 9-s − 0.632·10-s − 0.603·11-s − 1.15·12-s − 1.66·13-s − 1.03·15-s + 3/4·16-s + 0.970·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 0.426·22-s − 2.08·23-s + 0.408·24-s + 12/5·25-s + 1.17·26-s − 0.384·27-s + 8.91·29-s + 0.730·30-s − 0.718·31-s − 0.353·32-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(17318.0\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.431586502\)
\(L(\frac12)\) \(\approx\) \(3.431586502\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T )^{6} \)
good2 \( 1 + T - T^{2} - p T^{3} - p T^{4} + p^{3} T^{6} - p^{3} T^{8} - p^{4} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + 2 T + T^{2} - 2 T^{3} - 8 T^{4} - 2 p T^{5} + 7 T^{6} - 2 p^{2} T^{7} - 8 p^{2} T^{8} - 2 p^{3} T^{9} + p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T - 8 T^{2} + 12 T^{3} + 48 T^{4} - 26 T^{5} - 246 T^{6} - 26 p T^{7} + 48 p^{2} T^{8} + 12 p^{3} T^{9} - 8 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 23 T^{2} - 18 T^{3} + 360 T^{4} + 26 T^{5} - 4545 T^{6} + 26 p T^{7} + 360 p^{2} T^{8} - 18 p^{3} T^{9} - 23 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T - 25 T^{2} + 116 T^{3} + 424 T^{4} - 1320 T^{5} - 3775 T^{6} - 1320 p T^{7} + 424 p^{2} T^{8} + 116 p^{3} T^{9} - 25 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 4 T - 42 T^{2} - 64 T^{3} + 1670 T^{4} + 1212 T^{5} - 34070 T^{6} + 1212 p T^{7} + 1670 p^{2} T^{8} - 64 p^{3} T^{9} - 42 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 10 T + 30 T^{2} + 52 T^{3} + 50 T^{4} - 4230 T^{5} - 36290 T^{6} - 4230 p T^{7} + 50 p^{2} T^{8} + 52 p^{3} T^{9} + 30 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 4 T - 58 T^{2} - 232 T^{3} + 2126 T^{4} + 5028 T^{5} - 55854 T^{6} + 5028 p T^{7} + 2126 p^{2} T^{8} - 232 p^{3} T^{9} - 58 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 53 T^{2} - 248 T^{3} + 848 T^{4} + 6572 T^{5} + 12749 T^{6} + 6572 p T^{7} + 848 p^{2} T^{8} - 248 p^{3} T^{9} - 53 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 2 T + 95 T^{2} + 172 T^{3} + 95 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 8 T + 2 T^{2} + 80 T^{3} - 734 T^{4} - 16056 T^{5} - 71134 T^{6} - 16056 p T^{7} - 734 p^{2} T^{8} + 80 p^{3} T^{9} + 2 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 8 T - 60 T^{2} - 748 T^{3} + 1844 T^{4} + 25200 T^{5} + 59950 T^{6} + 25200 p T^{7} + 1844 p^{2} T^{8} - 748 p^{3} T^{9} - 60 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T - 5 T^{2} + 516 T^{3} + 66 T^{4} - 524 T^{5} + 375699 T^{6} - 524 p T^{7} + 66 p^{2} T^{8} + 516 p^{3} T^{9} - 5 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \)
67 \( 1 - 12 T + p T^{2} + 340 T^{3} - 6814 T^{4} + 21284 T^{5} - 42709 T^{6} + 21284 p T^{7} - 6814 p^{2} T^{8} + 340 p^{3} T^{9} + p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 6 T + 191 T^{2} + 868 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 10 T - 20 T^{2} - 1172 T^{3} - 6220 T^{4} + 20410 T^{5} + 600530 T^{6} + 20410 p T^{7} - 6220 p^{2} T^{8} - 1172 p^{3} T^{9} - 20 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 14 T - 46 T^{2} + 1004 T^{3} + 8702 T^{4} - 83502 T^{5} - 41298 T^{6} - 83502 p T^{7} + 8702 p^{2} T^{8} + 1004 p^{3} T^{9} - 46 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 2 T - 168 T^{2} - 476 T^{3} + 14408 T^{4} + 56742 T^{5} - 1250366 T^{6} + 56742 p T^{7} + 14408 p^{2} T^{8} - 476 p^{3} T^{9} - 168 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 10 T + 320 T^{2} - 1962 T^{3} + 320 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.78675076298302276214829733258, −5.46355984273638963438637246969, −5.31206813958636241939650218628, −5.18039264175491817948144228020, −4.84146415692001326045193405397, −4.75908726347504027279103993852, −4.70573741139571697393734194802, −4.57077906792325200306757563257, −4.53804492631684499752186101669, −4.21252836670046350555412295825, −4.15657259153233308153630103206, −3.44331498491179379180998787756, −3.30609991253543062908775183409, −3.29570918993894927309806988445, −2.98731211646972824621625050261, −2.84483254243896179756225455504, −2.70773884382702482345612865720, −2.22097348516636377598838239691, −2.18656468621651464154888870987, −2.13434543047120394955402936252, −1.75351923697494255766619066191, −1.17963907157478417566298123936, −0.983812211928118496442745611831, −0.833195135585768240361321609726, −0.56694498645074791712338353965, 0.56694498645074791712338353965, 0.833195135585768240361321609726, 0.983812211928118496442745611831, 1.17963907157478417566298123936, 1.75351923697494255766619066191, 2.13434543047120394955402936252, 2.18656468621651464154888870987, 2.22097348516636377598838239691, 2.70773884382702482345612865720, 2.84483254243896179756225455504, 2.98731211646972824621625050261, 3.29570918993894927309806988445, 3.30609991253543062908775183409, 3.44331498491179379180998787756, 4.15657259153233308153630103206, 4.21252836670046350555412295825, 4.53804492631684499752186101669, 4.57077906792325200306757563257, 4.70573741139571697393734194802, 4.75908726347504027279103993852, 4.84146415692001326045193405397, 5.18039264175491817948144228020, 5.31206813958636241939650218628, 5.46355984273638963438637246969, 5.78675076298302276214829733258

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

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