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Properties

Label 12-5292e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.196\times 10^{22}$
Sign $1$
Analytic cond. $5.69353\times 10^{9}$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·11-s + 3·13-s − 2·17-s + 3·19-s + 14·23-s + 11·25-s + 29-s + 6·31-s + 3·37-s − 3·43-s + 42·47-s + 6·53-s − 2·55-s + 62·59-s − 12·61-s − 3·65-s + 12·67-s − 34·71-s − 3·73-s − 18·79-s − 20·83-s + 2·85-s − 12·89-s − 3·95-s − 9·97-s − 13·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.603·11-s + 0.832·13-s − 0.485·17-s + 0.688·19-s + 2.91·23-s + 11/5·25-s + 0.185·29-s + 1.07·31-s + 0.493·37-s − 0.457·43-s + 6.12·47-s + 0.824·53-s − 0.269·55-s + 8.07·59-s − 1.53·61-s − 0.372·65-s + 1.46·67-s − 4.03·71-s − 0.351·73-s − 2.02·79-s − 2.19·83-s + 0.216·85-s − 1.27·89-s − 0.307·95-s − 0.913·97-s − 1.29·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(5.69353\times 10^{9}\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5330797632\)
\(L(\frac12)\) \(\approx\) \(0.5330797632\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T - 2 p T^{2} - 7 T^{3} + 57 T^{4} + 14 T^{5} - 299 T^{6} + 14 p T^{7} + 57 p^{2} T^{8} - 7 p^{3} T^{9} - 2 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T - 4 T^{2} - 46 T^{3} + 6 T^{4} + 230 T^{5} + 1699 T^{6} + 230 p T^{7} + 6 p^{2} T^{8} - 46 p^{3} T^{9} - 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 3 T^{2} + 84 T^{3} - 15 p T^{4} - 345 T^{5} + 5006 T^{6} - 345 p T^{7} - 15 p^{3} T^{8} + 84 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T - 28 T^{2} + 22 T^{3} + 438 T^{4} - 926 T^{5} - 8297 T^{6} - 926 p T^{7} + 438 p^{2} T^{8} + 22 p^{3} T^{9} - 28 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 24 T^{2} - 29 T^{3} + 357 T^{4} + 1524 T^{5} - 8997 T^{6} + 1524 p T^{7} + 357 p^{2} T^{8} - 29 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 14 T + 74 T^{2} - 358 T^{3} + 2628 T^{4} - 11188 T^{5} + 33943 T^{6} - 11188 p T^{7} + 2628 p^{2} T^{8} - 358 p^{3} T^{9} + 74 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - T - 46 T^{2} - 149 T^{3} + 897 T^{4} + 4282 T^{5} - 13523 T^{6} + 4282 p T^{7} + 897 p^{2} T^{8} - 149 p^{3} T^{9} - 46 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 3 T + 57 T^{2} - 159 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 3 T - 72 T^{2} + 155 T^{3} + 2967 T^{4} - 2244 T^{5} - 114171 T^{6} - 2244 p T^{7} + 2967 p^{2} T^{8} + 155 p^{3} T^{9} - 72 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 810 p T^{7} + 4410 p^{2} T^{8} + 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 14820 p T^{7} - 1947 p^{2} T^{8} - 979 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 21 T + 261 T^{2} - 2181 T^{3} + 261 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 6 T - 126 T^{2} + 282 T^{3} + 13896 T^{4} - 15396 T^{5} - 801173 T^{6} - 15396 p T^{7} + 13896 p^{2} T^{8} + 282 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 31 T + 485 T^{2} - 4647 T^{3} + 485 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 6 T - 12 T^{2} - 357 T^{3} - 12 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 6 T + 186 T^{2} - 811 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 17 T + 119 T^{2} + 507 T^{3} + 119 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 3 T - 186 T^{2} - 133 T^{3} + 22713 T^{4} + 582 T^{5} - 1916871 T^{6} + 582 p T^{7} + 22713 p^{2} T^{8} - 133 p^{3} T^{9} - 186 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 9 T + 195 T^{2} + 1053 T^{3} + 195 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 20 T + 38 T^{2} + 346 T^{3} + 32058 T^{4} + 183754 T^{5} - 606869 T^{6} + 183754 p T^{7} + 32058 p^{2} T^{8} + 346 p^{3} T^{9} + 38 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T - 72 T^{2} - 258 T^{3} + 10332 T^{4} - 58524 T^{5} - 1852445 T^{6} - 58524 p T^{7} + 10332 p^{2} T^{8} - 258 p^{3} T^{9} - 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 9 T - 66 T^{2} - 2023 T^{3} - 7707 T^{4} + 73950 T^{5} + 1766073 T^{6} + 73950 p T^{7} - 7707 p^{2} T^{8} - 2023 p^{3} T^{9} - 66 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.14124555687336704352872391466, −4.00972488063878283382021516272, −3.94704638453354656606083358771, −3.90409508568202415761013408896, −3.69599644146345155429053505333, −3.57256893721706707095463701911, −3.17679210052322044946105783517, −3.17494736124709141206622677122, −3.08384311835473557406475616349, −2.81058146569965841526291491181, −2.80035889054993851431617375727, −2.65162272781983801955602011845, −2.59663354383671825979537453545, −2.41048228313046925994644427401, −2.16232760142016531584144585903, −2.03001042514844522596943406314, −1.86292110275334956113603812168, −1.51628835634084554840869990851, −1.17990430847891543816193802704, −1.16461036875003957458652084660, −1.10190249055960554256725067079, −0.821055696294594162700270925097, −0.76906473616783038635494682148, −0.76628959441706068106156903817, −0.04409564765249422588542056230, 0.04409564765249422588542056230, 0.76628959441706068106156903817, 0.76906473616783038635494682148, 0.821055696294594162700270925097, 1.10190249055960554256725067079, 1.16461036875003957458652084660, 1.17990430847891543816193802704, 1.51628835634084554840869990851, 1.86292110275334956113603812168, 2.03001042514844522596943406314, 2.16232760142016531584144585903, 2.41048228313046925994644427401, 2.59663354383671825979537453545, 2.65162272781983801955602011845, 2.80035889054993851431617375727, 2.81058146569965841526291491181, 3.08384311835473557406475616349, 3.17494736124709141206622677122, 3.17679210052322044946105783517, 3.57256893721706707095463701911, 3.69599644146345155429053505333, 3.90409508568202415761013408896, 3.94704638453354656606083358771, 4.00972488063878283382021516272, 4.14124555687336704352872391466

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

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