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Properties

Label 12-9280e6-1.1-c1e6-0-2
Degree $12$
Conductor $6.387\times 10^{23}$
Sign $1$
Analytic cond. $1.65558\times 10^{11}$
Root an. cond. $8.60820$
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  − 3-s + 6·5-s + 3·7-s − 8·9-s + 4·11-s − 9·13-s − 6·15-s − 11·17-s − 8·19-s − 3·21-s + 23-s + 21·25-s + 8·27-s + 6·29-s + 3·31-s − 4·33-s + 18·35-s − 14·37-s + 9·39-s − 2·41-s − 23·43-s − 48·45-s − 2·47-s − 14·49-s + 11·51-s − 15·53-s + 24·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 2.68·5-s + 1.13·7-s − 8/3·9-s + 1.20·11-s − 2.49·13-s − 1.54·15-s − 2.66·17-s − 1.83·19-s − 0.654·21-s + 0.208·23-s + 21/5·25-s + 1.53·27-s + 1.11·29-s + 0.538·31-s − 0.696·33-s + 3.04·35-s − 2.30·37-s + 1.44·39-s − 0.312·41-s − 3.50·43-s − 7.15·45-s − 0.291·47-s − 2·49-s + 1.54·51-s − 2.06·53-s + 3.23·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{36} \cdot 5^{6} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(1.65558\times 10^{11}\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{36} \cdot 5^{6} \cdot 29^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
5 \( ( 1 - T )^{6} \)
29 \( ( 1 - T )^{6} \)
good3 \( 1 + T + p^{2} T^{2} + p^{2} T^{3} + 13 p T^{4} + 40 T^{5} + 122 T^{6} + 40 p T^{7} + 13 p^{3} T^{8} + p^{5} T^{9} + p^{6} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T + 23 T^{2} - 39 T^{3} + 135 T^{4} - 8 p T^{5} + 318 T^{6} - 8 p^{2} T^{7} + 135 p^{2} T^{8} - 39 p^{3} T^{9} + 23 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T + 50 T^{2} - 156 T^{3} + 1131 T^{4} - 2848 T^{5} + 15428 T^{6} - 2848 p T^{7} + 1131 p^{2} T^{8} - 156 p^{3} T^{9} + 50 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 9 T + 7 p T^{2} + 501 T^{3} + 2931 T^{4} + 902 p T^{5} + 49874 T^{6} + 902 p^{2} T^{7} + 2931 p^{2} T^{8} + 501 p^{3} T^{9} + 7 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 11 T + 87 T^{2} + 507 T^{3} + 2871 T^{4} + 14114 T^{5} + 63590 T^{6} + 14114 p T^{7} + 2871 p^{2} T^{8} + 507 p^{3} T^{9} + 87 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 8 T + 118 T^{2} + 696 T^{3} + 5659 T^{4} + 25320 T^{5} + 143580 T^{6} + 25320 p T^{7} + 5659 p^{2} T^{8} + 696 p^{3} T^{9} + 118 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - T + 73 T^{2} - 61 T^{3} + 2851 T^{4} - 1248 T^{5} + 76474 T^{6} - 1248 p T^{7} + 2851 p^{2} T^{8} - 61 p^{3} T^{9} + 73 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T + 65 T^{2} - 147 T^{3} + 3347 T^{4} - 8124 T^{5} + 121674 T^{6} - 8124 p T^{7} + 3347 p^{2} T^{8} - 147 p^{3} T^{9} + 65 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 14 T + 6 p T^{2} + 1890 T^{3} + 17811 T^{4} + 113104 T^{5} + 817356 T^{6} + 113104 p T^{7} + 17811 p^{2} T^{8} + 1890 p^{3} T^{9} + 6 p^{5} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T + 66 T^{2} + 330 T^{3} + 5167 T^{4} + 22116 T^{5} + 195900 T^{6} + 22116 p T^{7} + 5167 p^{2} T^{8} + 330 p^{3} T^{9} + 66 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 23 T + 315 T^{2} + 2699 T^{3} + 373 p T^{4} + 64972 T^{5} + 297950 T^{6} + 64972 p T^{7} + 373 p^{3} T^{8} + 2699 p^{3} T^{9} + 315 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 2 T + 130 T^{2} + 62 T^{3} + 9323 T^{4} + 1004 T^{5} + 508580 T^{6} + 1004 p T^{7} + 9323 p^{2} T^{8} + 62 p^{3} T^{9} + 130 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 15 T + 139 T^{2} + 655 T^{3} + 3443 T^{4} + 14798 T^{5} + 158706 T^{6} + 14798 p T^{7} + 3443 p^{2} T^{8} + 655 p^{3} T^{9} + 139 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + T + 249 T^{2} + 287 T^{3} + 30259 T^{4} + 32962 T^{5} + 2227734 T^{6} + 32962 p T^{7} + 30259 p^{2} T^{8} + 287 p^{3} T^{9} + 249 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 5 T + 165 T^{2} + 197 T^{3} + 15227 T^{4} + 28514 T^{5} + 1259374 T^{6} + 28514 p T^{7} + 15227 p^{2} T^{8} + 197 p^{3} T^{9} + 165 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 26 T + 438 T^{2} + 3918 T^{3} + 19515 T^{4} - 58436 T^{5} - 1147380 T^{6} - 58436 p T^{7} + 19515 p^{2} T^{8} + 3918 p^{3} T^{9} + 438 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T + 306 T^{2} + 3524 T^{3} + 46927 T^{4} + 444888 T^{5} + 4282332 T^{6} + 444888 p T^{7} + 46927 p^{2} T^{8} + 3524 p^{3} T^{9} + 306 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 19 T + 407 T^{2} + 4783 T^{3} + 63395 T^{4} + 564502 T^{5} + 5735566 T^{6} + 564502 p T^{7} + 63395 p^{2} T^{8} + 4783 p^{3} T^{9} + 407 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 33 T + 643 T^{2} - 9101 T^{3} + 108623 T^{4} - 1142944 T^{5} + 10813862 T^{6} - 1142944 p T^{7} + 108623 p^{2} T^{8} - 9101 p^{3} T^{9} + 643 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 4 T + 258 T^{2} + 252 T^{3} + 25499 T^{4} + 158456 T^{5} + 1897636 T^{6} + 158456 p T^{7} + 25499 p^{2} T^{8} + 252 p^{3} T^{9} + 258 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 18 T + 266 T^{2} + 2570 T^{3} + 33151 T^{4} + 315796 T^{5} + 3525452 T^{6} + 315796 p T^{7} + 33151 p^{2} T^{8} + 2570 p^{3} T^{9} + 266 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 21 T + 591 T^{2} + 8397 T^{3} + 139955 T^{4} + 1492686 T^{5} + 17914030 T^{6} + 1492686 p T^{7} + 139955 p^{2} T^{8} + 8397 p^{3} T^{9} + 591 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.41099087437111458199090685371, −4.27879145273929162699123448205, −4.12247946790300697696464742851, −3.79792449812926301329636367460, −3.77432847136671810645423402728, −3.67567580934984631625313006455, −3.57349054071406219264405753409, −3.10197314841421876765125040688, −2.99150530449686154799276915172, −2.98226636083617771033081472734, −2.98030993176865692056435130121, −2.94495041903708258810910717525, −2.80797366940702909173799719100, −2.38526374706158064518798204455, −2.27395269429369578408026066870, −2.23744635631973622400115090226, −2.12521389445413617213500804138, −2.03082598957250004838893325842, −1.90116656011563494908397108767, −1.71003269604296931836053639807, −1.54515832549501983664172580995, −1.33754892114923026926951651326, −1.25352740469002041180276189095, −1.10151255223124431345849598360, −1.00473306938680593794989116524, 0, 0, 0, 0, 0, 0, 1.00473306938680593794989116524, 1.10151255223124431345849598360, 1.25352740469002041180276189095, 1.33754892114923026926951651326, 1.54515832549501983664172580995, 1.71003269604296931836053639807, 1.90116656011563494908397108767, 2.03082598957250004838893325842, 2.12521389445413617213500804138, 2.23744635631973622400115090226, 2.27395269429369578408026066870, 2.38526374706158064518798204455, 2.80797366940702909173799719100, 2.94495041903708258810910717525, 2.98030993176865692056435130121, 2.98226636083617771033081472734, 2.99150530449686154799276915172, 3.10197314841421876765125040688, 3.57349054071406219264405753409, 3.67567580934984631625313006455, 3.77432847136671810645423402728, 3.79792449812926301329636367460, 4.12247946790300697696464742851, 4.27879145273929162699123448205, 4.41099087437111458199090685371

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

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