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Properties

Label 12-8280e6-1.1-c1e6-0-1
Degree $12$
Conductor $3.222\times 10^{23}$
Sign $1$
Analytic cond. $8.35304\times 10^{10}$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 2·7-s + 4·11-s − 4·17-s + 8·19-s − 6·23-s + 21·25-s − 18·29-s − 8·31-s + 12·35-s + 6·37-s − 6·41-s + 8·43-s + 8·47-s − 14·49-s − 8·53-s − 24·55-s − 2·59-s − 8·61-s − 2·67-s − 2·71-s + 8·73-s − 8·77-s − 28·79-s − 24·83-s + 24·85-s − 4·89-s + ⋯
L(s)  = 1  − 2.68·5-s − 0.755·7-s + 1.20·11-s − 0.970·17-s + 1.83·19-s − 1.25·23-s + 21/5·25-s − 3.34·29-s − 1.43·31-s + 2.02·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 2·49-s − 1.09·53-s − 3.23·55-s − 0.260·59-s − 1.02·61-s − 0.244·67-s − 0.237·71-s + 0.936·73-s − 0.911·77-s − 3.15·79-s − 2.63·83-s + 2.60·85-s − 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(8.35304\times 10^{10}\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( ( 1 + T )^{6} \)
23 \( ( 1 + T )^{6} \)
good7 \( 1 + 2 T + 18 T^{2} + 40 T^{3} + 234 T^{4} + 66 p T^{5} + 1842 T^{6} + 66 p^{2} T^{7} + 234 p^{2} T^{8} + 40 p^{3} T^{9} + 18 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T + 28 T^{2} - 120 T^{3} + 507 T^{4} - 1700 T^{5} + 7056 T^{6} - 1700 p T^{7} + 507 p^{2} T^{8} - 120 p^{3} T^{9} + 28 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} - 60 T^{3} + 411 T^{4} - 2292 T^{5} + 2880 T^{6} - 2292 p T^{7} + 411 p^{2} T^{8} - 60 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 4 T + 30 T^{2} + 64 T^{3} + 678 T^{4} + 1332 T^{5} + 12158 T^{6} + 1332 p T^{7} + 678 p^{2} T^{8} + 64 p^{3} T^{9} + 30 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 8 T + 96 T^{2} - 556 T^{3} + 4019 T^{4} - 18404 T^{5} + 98408 T^{6} - 18404 p T^{7} + 4019 p^{2} T^{8} - 556 p^{3} T^{9} + 96 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 18 T + 164 T^{2} + 772 T^{3} + 1124 T^{4} - 13454 T^{5} - 112530 T^{6} - 13454 p T^{7} + 1124 p^{2} T^{8} + 772 p^{3} T^{9} + 164 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 152 T^{2} + 832 T^{3} + 9816 T^{4} + 42568 T^{5} + 383406 T^{6} + 42568 p T^{7} + 9816 p^{2} T^{8} + 832 p^{3} T^{9} + 152 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 6 T + 66 T^{2} - 72 T^{3} + 2058 T^{4} - 150 T^{5} + 81618 T^{6} - 150 p T^{7} + 2058 p^{2} T^{8} - 72 p^{3} T^{9} + 66 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 120 T^{2} + 636 T^{3} + 6324 T^{4} + 30510 T^{5} + 248138 T^{6} + 30510 p T^{7} + 6324 p^{2} T^{8} + 636 p^{3} T^{9} + 120 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 214 T^{2} - 1416 T^{3} + 20631 T^{4} - 110816 T^{5} + 1143316 T^{6} - 110816 p T^{7} + 20631 p^{2} T^{8} - 1416 p^{3} T^{9} + 214 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 8 T + 160 T^{2} - 756 T^{3} + 10275 T^{4} - 33724 T^{5} + 489144 T^{6} - 33724 p T^{7} + 10275 p^{2} T^{8} - 756 p^{3} T^{9} + 160 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 8 T + 230 T^{2} + 1412 T^{3} + 24878 T^{4} + 124672 T^{5} + 1651414 T^{6} + 124672 p T^{7} + 24878 p^{2} T^{8} + 1412 p^{3} T^{9} + 230 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T + 296 T^{2} + 612 T^{3} + 39164 T^{4} + 72854 T^{5} + 2970962 T^{6} + 72854 p T^{7} + 39164 p^{2} T^{8} + 612 p^{3} T^{9} + 296 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 8 T + 224 T^{2} + 812 T^{3} + 18411 T^{4} + 14780 T^{5} + 1072984 T^{6} + 14780 p T^{7} + 18411 p^{2} T^{8} + 812 p^{3} T^{9} + 224 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 270 T^{2} + 796 T^{3} + 33714 T^{4} + 116958 T^{5} + 2694942 T^{6} + 116958 p T^{7} + 33714 p^{2} T^{8} + 796 p^{3} T^{9} + 270 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 2 T + 252 T^{2} + 940 T^{3} + 33916 T^{4} + 117734 T^{5} + 3025574 T^{6} + 117734 p T^{7} + 33916 p^{2} T^{8} + 940 p^{3} T^{9} + 252 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T + 200 T^{2} - 1372 T^{3} + 25899 T^{4} - 149452 T^{5} + 2122008 T^{6} - 149452 p T^{7} + 25899 p^{2} T^{8} - 1372 p^{3} T^{9} + 200 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 28 T + 626 T^{2} + 9332 T^{3} + 122319 T^{4} + 1280600 T^{5} + 12451644 T^{6} + 1280600 p T^{7} + 122319 p^{2} T^{8} + 9332 p^{3} T^{9} + 626 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 24 T + 382 T^{2} + 3180 T^{3} + 21614 T^{4} + 768 p T^{5} + 548518 T^{6} + 768 p^{2} T^{7} + 21614 p^{2} T^{8} + 3180 p^{3} T^{9} + 382 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 4 T + 262 T^{2} + 1428 T^{3} + 43551 T^{4} + 210056 T^{5} + 4643796 T^{6} + 210056 p T^{7} + 43551 p^{2} T^{8} + 1428 p^{3} T^{9} + 262 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 24 T + 494 T^{2} - 6904 T^{3} + 85743 T^{4} - 915568 T^{5} + 9189188 T^{6} - 915568 p T^{7} + 85743 p^{2} T^{8} - 6904 p^{3} T^{9} + 494 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.39934338544040841365853623453, −4.10240809599903416806967830271, −4.08154547894327224074624919613, −3.91185499272339669213848071974, −3.76246248583278680759670205321, −3.73089460353128114033543214228, −3.66343183409194867610636933717, −3.35643656743831671589264830673, −3.24806182274294656724547292281, −3.23486969686926255621029979158, −3.20809848707366410967214863805, −3.20301535088844592094083587369, −2.77773608075863997609405492733, −2.52209972339267570649971364319, −2.41756335029623059927538111647, −2.29082732975047490395751587151, −2.23188509126408112194987996076, −2.12647068403569762121825206823, −1.93266073219940448229579817567, −1.47657294554963078515004971132, −1.31667163618586958214738527179, −1.17773040801798133428578262269, −1.16478462520119766240477514849, −1.15135954463553972786439346373, −1.05524528948682544336768653046, 0, 0, 0, 0, 0, 0, 1.05524528948682544336768653046, 1.15135954463553972786439346373, 1.16478462520119766240477514849, 1.17773040801798133428578262269, 1.31667163618586958214738527179, 1.47657294554963078515004971132, 1.93266073219940448229579817567, 2.12647068403569762121825206823, 2.23188509126408112194987996076, 2.29082732975047490395751587151, 2.41756335029623059927538111647, 2.52209972339267570649971364319, 2.77773608075863997609405492733, 3.20301535088844592094083587369, 3.20809848707366410967214863805, 3.23486969686926255621029979158, 3.24806182274294656724547292281, 3.35643656743831671589264830673, 3.66343183409194867610636933717, 3.73089460353128114033543214228, 3.76246248583278680759670205321, 3.91185499272339669213848071974, 4.08154547894327224074624919613, 4.10240809599903416806967830271, 4.39934338544040841365853623453

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

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