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Properties

Label 12-58e6-1.1-c1e6-0-0
Degree $12$
Conductor $38068692544$
Sign $1$
Analytic cond. $0.00986803$
Root an. cond. $0.680538$
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 + 3·3-s − 4·5-s + 3·6-s − 5·7-s + 3·9-s − 4·10-s − 6·11-s + 11·13-s − 5·14-s − 12·15-s − 10·17-s + 3·18-s − 6·19-s − 15·21-s − 6·22-s − 7·23-s + 19·25-s + 11·26-s + 7·27-s + 15·29-s − 12·30-s + 4·31-s − 18·33-s − 10·34-s + 20·35-s − 13·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1.78·5-s + 1.22·6-s − 1.88·7-s + 9-s − 1.26·10-s − 1.80·11-s + 3.05·13-s − 1.33·14-s − 3.09·15-s − 2.42·17-s + 0.707·18-s − 1.37·19-s − 3.27·21-s − 1.27·22-s − 1.45·23-s + 19/5·25-s + 2.15·26-s + 1.34·27-s + 2.78·29-s − 2.19·30-s + 0.718·31-s − 3.13·33-s − 1.71·34-s + 3.38·35-s − 2.13·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(0.00986803\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 29^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6009044717\)
\(L(\frac12)\) \(\approx\) \(0.6009044717\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
29 \( 1 - 15 T + 113 T^{2} - 659 T^{3} + 113 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 - p T + 2 p T^{2} - 16 T^{3} + 44 T^{4} - 77 T^{5} + 127 T^{6} - 77 p T^{7} + 44 p^{2} T^{8} - 16 p^{3} T^{9} + 2 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + 4 T - 3 T^{2} - 46 T^{3} - 17 p T^{4} + 142 T^{5} + 811 T^{6} + 142 p T^{7} - 17 p^{3} T^{8} - 46 p^{3} T^{9} - 3 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 5 T + 18 T^{2} + 83 T^{3} + 303 T^{4} + 878 T^{5} + 2381 T^{6} + 878 p T^{7} + 303 p^{2} T^{8} + 83 p^{3} T^{9} + 18 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T + 25 T^{2} + 84 T^{3} + 355 T^{4} + 1500 T^{5} + 6229 T^{6} + 1500 p T^{7} + 355 p^{2} T^{8} + 84 p^{3} T^{9} + 25 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 11 T + 45 T^{2} - 5 p T^{3} - 18 p T^{4} + 2656 T^{5} - 13147 T^{6} + 2656 p T^{7} - 18 p^{3} T^{8} - 5 p^{4} T^{9} + 45 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 5 T + 43 T^{2} + 129 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 + 6 T + 2 p T^{2} + 191 T^{3} + 851 T^{4} + 3745 T^{5} + 9927 T^{6} + 3745 p T^{7} + 851 p^{2} T^{8} + 191 p^{3} T^{9} + 2 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T + 61 T^{2} + 21 p T^{3} + 2790 T^{4} + 16156 T^{5} + 79995 T^{6} + 16156 p T^{7} + 2790 p^{2} T^{8} + 21 p^{4} T^{9} + 61 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 4 T + 55 T^{2} - 376 T^{3} + 3299 T^{4} - 16450 T^{5} + 115991 T^{6} - 16450 p T^{7} + 3299 p^{2} T^{8} - 376 p^{3} T^{9} + 55 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 13 T + 62 T^{2} + 10 T^{3} - 239 T^{4} + 4489 T^{5} + 54656 T^{6} + 4489 p T^{7} - 239 p^{2} T^{8} + 10 p^{3} T^{9} + 62 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 6 T + 86 T^{2} + 493 T^{3} + 86 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 7 T + 48 T^{2} - 28 T^{3} - 349 T^{4} - 2919 T^{5} + 51400 T^{6} - 2919 p T^{7} - 349 p^{2} T^{8} - 28 p^{3} T^{9} + 48 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T - 38 T^{2} - 60 T^{3} + 2596 T^{4} - 5269 T^{5} - 76609 T^{6} - 5269 p T^{7} + 2596 p^{2} T^{8} - 60 p^{3} T^{9} - 38 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 22 T + 158 T^{2} - 63 T^{3} - 4279 T^{4} + 42971 T^{5} + 747025 T^{6} + 42971 p T^{7} - 4279 p^{2} T^{8} - 63 p^{3} T^{9} + 158 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 19 T + 281 T^{2} + 2411 T^{3} + 281 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( 1 - 31 T + 361 T^{2} - 1705 T^{3} - 3718 T^{4} + 140576 T^{5} - 1456267 T^{6} + 140576 p T^{7} - 3718 p^{2} T^{8} - 1705 p^{3} T^{9} + 361 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 11 T + 54 T^{2} + 143 T^{3} - 5191 T^{4} + 47520 T^{5} - 174923 T^{6} + 47520 p T^{7} - 5191 p^{2} T^{8} + 143 p^{3} T^{9} + 54 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 23 T + 248 T^{2} - 2608 T^{3} + 25807 T^{4} - 189181 T^{5} + 1381968 T^{6} - 189181 p T^{7} + 25807 p^{2} T^{8} - 2608 p^{3} T^{9} + 248 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 5 T - 48 T^{2} + 1676 T^{3} - 5590 T^{4} - 56199 T^{5} + 1395211 T^{6} - 56199 p T^{7} - 5590 p^{2} T^{8} + 1676 p^{3} T^{9} - 48 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 23 T^{2} - 112 T^{3} + 4813 T^{4} + 32998 T^{5} - 315155 T^{6} + 32998 p T^{7} + 4813 p^{2} T^{8} - 112 p^{3} T^{9} - 23 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 - 35 T + 547 T^{2} - 6153 T^{3} + 70526 T^{4} - 784756 T^{5} + 7621055 T^{6} - 784756 p T^{7} + 70526 p^{2} T^{8} - 6153 p^{3} T^{9} + 547 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 13 T + 80 T^{2} + 61 T^{3} + 2797 T^{4} - 36470 T^{5} + 591921 T^{6} - 36470 p T^{7} + 2797 p^{2} T^{8} + 61 p^{3} T^{9} + 80 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 20 T + 51 T^{2} + 2026 T^{3} - 17103 T^{4} + 1492 T^{5} + 727355 T^{6} + 1492 p T^{7} - 17103 p^{2} T^{8} + 2026 p^{3} T^{9} + 51 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
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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

−8.773093694927719344852321470889, −8.510873813681926101157526716007, −8.225239511846850158540244353264, −8.107000450416913811443947421351, −8.086854333327744959624170888822, −8.066082294227148422261423416046, −7.67839627775946668218558221931, −6.77376692475317745727957685680, −6.72739288050964183111280694597, −6.64979648222450654618329047919, −6.58549716440491780741281232975, −6.56402969572114070863579091511, −6.22411541886194848638701809957, −5.47263143702377533155607807990, −5.32943593020712940336867539766, −4.91637049011303650429150678343, −4.51837452040187098641598730832, −4.42542696393205624443681899490, −4.28824881609124735563462457205, −3.49357046997190486475942377954, −3.42700433531875783992323969808, −3.25524393388551810454495766471, −3.16637332030411344290933635322, −2.51267149914475230714086821315, −2.10258903573578906270884423837, 2.10258903573578906270884423837, 2.51267149914475230714086821315, 3.16637332030411344290933635322, 3.25524393388551810454495766471, 3.42700433531875783992323969808, 3.49357046997190486475942377954, 4.28824881609124735563462457205, 4.42542696393205624443681899490, 4.51837452040187098641598730832, 4.91637049011303650429150678343, 5.32943593020712940336867539766, 5.47263143702377533155607807990, 6.22411541886194848638701809957, 6.56402969572114070863579091511, 6.58549716440491780741281232975, 6.64979648222450654618329047919, 6.72739288050964183111280694597, 6.77376692475317745727957685680, 7.67839627775946668218558221931, 8.066082294227148422261423416046, 8.086854333327744959624170888822, 8.107000450416913811443947421351, 8.225239511846850158540244353264, 8.510873813681926101157526716007, 8.773093694927719344852321470889

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

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