Properties

Label 2-97461-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 2·5-s − 4·10-s − 5·11-s − 13-s − 4·16-s + 17-s − 6·19-s + 4·20-s + 10·22-s + 4·23-s − 25-s + 2·26-s − 8·29-s + 31-s + 8·32-s − 2·34-s + 37-s + 12·38-s + 3·41-s + 11·43-s − 10·44-s − 8·46-s − 2·47-s + 2·50-s − 2·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s − 1.50·11-s − 0.277·13-s − 16-s + 0.242·17-s − 1.37·19-s + 0.894·20-s + 2.13·22-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 1.48·29-s + 0.179·31-s + 1.41·32-s − 0.342·34-s + 0.164·37-s + 1.94·38-s + 0.468·41-s + 1.67·43-s − 1.50·44-s − 1.17·46-s − 0.291·47-s + 0.282·50-s − 0.277·52-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: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5051481412\)
\(L(\frac12)\) \(\approx\) \(0.5051481412\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.68456268469901, −13.14867436935761, −12.91683076865858, −12.44966127537796, −11.46335707765627, −11.14229559236737, −10.60270466496379, −10.25172942101668, −9.866506420517052, −9.202648299527852, −9.025755170619749, −8.236979114603244, −7.959753268259698, −7.300654449447281, −7.001578800541453, −6.172492269393290, −5.690717662106299, −5.211962500472691, −4.498726136675276, −3.906950065731506, −2.818416510863091, −2.408517233193856, −1.928543348430455, −1.177396749153840, −0.2946724267712450, 0.2946724267712450, 1.177396749153840, 1.928543348430455, 2.408517233193856, 2.818416510863091, 3.906950065731506, 4.498726136675276, 5.211962500472691, 5.690717662106299, 6.172492269393290, 7.001578800541453, 7.300654449447281, 7.959753268259698, 8.236979114603244, 9.025755170619749, 9.202648299527852, 9.866506420517052, 10.25172942101668, 10.60270466496379, 11.14229559236737, 11.46335707765627, 12.44966127537796, 12.91683076865858, 13.14867436935761, 13.68456268469901

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