Properties

Label 2-94136-1.1-c1-0-6
Degree $2$
Conductor $94136$
Sign $-1$
Analytic cond. $751.679$
Root an. cond. $27.4167$
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  + 2·5-s + 7-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s + 2·35-s − 2·37-s − 4·43-s − 6·45-s + 8·47-s + 49-s − 6·53-s + 8·55-s − 6·61-s − 3·63-s − 4·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s − 16·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 0.768·61-s − 0.377·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(94136\)    =    \(2^{3} \cdot 7 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(751.679\)
Root analytic conductor: \(27.4167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{94136} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 94136,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 - T \)
41 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.03470536682208, −13.80072593968190, −13.12958503863315, −12.45766766126621, −12.14040598349142, −11.68548853075618, −11.06925948755158, −10.66451453028987, −9.963089331167597, −9.694672724865401, −9.045307577793577, −8.675977776692689, −8.053667357798480, −7.652631274583447, −6.812852378979854, −6.319245205697315, −5.979443054579571, −5.377976816865547, −4.893256560765979, −4.115999803744430, −3.643984012718271, −2.857977177832026, −2.245247711153567, −1.717900888624591, −0.9880469692498401, 0, 0.9880469692498401, 1.717900888624591, 2.245247711153567, 2.857977177832026, 3.643984012718271, 4.115999803744430, 4.893256560765979, 5.377976816865547, 5.979443054579571, 6.319245205697315, 6.812852378979854, 7.652631274583447, 8.053667357798480, 8.675977776692689, 9.045307577793577, 9.694672724865401, 9.963089331167597, 10.66451453028987, 11.06925948755158, 11.68548853075618, 12.14040598349142, 12.45766766126621, 13.12958503863315, 13.80072593968190, 14.03470536682208

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