Properties

Label 2-97461-1.1-c1-0-24
Degree $2$
Conductor $97461$
Sign $-1$
Analytic cond. $778.230$
Root an. cond. $27.8967$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s + 6·11-s + 13-s − 16-s + 17-s − 4·19-s − 2·20-s − 6·22-s − 6·23-s − 25-s − 26-s + 6·29-s + 2·31-s − 5·32-s − 34-s + 2·37-s + 4·38-s + 6·40-s − 6·41-s − 6·44-s + 6·46-s − 4·47-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s + 1.80·11-s + 0.277·13-s − 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.447·20-s − 1.27·22-s − 1.25·23-s − 1/5·25-s − 0.196·26-s + 1.11·29-s + 0.359·31-s − 0.883·32-s − 0.171·34-s + 0.328·37-s + 0.648·38-s + 0.948·40-s − 0.937·41-s − 0.904·44-s + 0.884·46-s − 0.583·47-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(778.230\)
Root analytic conductor: \(27.8967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97461} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.95519141410564, −13.66157868951961, −13.21036596913273, −12.41465764819961, −12.18055154578874, −11.52059330519823, −10.95416714372816, −10.40672689318627, −9.899157249251742, −9.551366561261584, −9.211425215696145, −8.554426468337395, −8.195592451299666, −7.746171135640352, −6.676601685459218, −6.594110568712284, −6.035206763889542, −5.349089961825709, −4.690423319593169, −4.109249362430988, −3.762667906155529, −2.884319136471102, −1.919499257033144, −1.599546645440323, −0.9466098330543203, 0, 0.9466098330543203, 1.599546645440323, 1.919499257033144, 2.884319136471102, 3.762667906155529, 4.109249362430988, 4.690423319593169, 5.349089961825709, 6.035206763889542, 6.594110568712284, 6.676601685459218, 7.746171135640352, 8.195592451299666, 8.554426468337395, 9.211425215696145, 9.551366561261584, 9.899157249251742, 10.40672689318627, 10.95416714372816, 11.52059330519823, 12.18055154578874, 12.41465764819961, 13.21036596913273, 13.66157868951961, 13.95519141410564

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