Properties

Label 2-94192-1.1-c1-0-13
Degree $2$
Conductor $94192$
Sign $1$
Analytic cond. $752.126$
Root an. cond. $27.4249$
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  + 2·5-s + 7-s − 3·9-s − 4·11-s + 2·13-s + 6·17-s + 8·19-s − 25-s + 8·31-s + 2·35-s + 2·37-s − 2·41-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 + 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.312·41-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(94192\)    =    \(2^{4} \cdot 7 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(752.126\)
Root analytic conductor: \(27.4249\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{94192} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 94192,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.411166104\)
\(L(\frac12)\) \(\approx\) \(3.411166104\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
29 \( 1 \)
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 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
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

−13.89870766194053, −13.38743470588570, −13.05479452764172, −12.23508715093456, −11.77622181925723, −11.48742092078936, −10.84106350418483, −10.21270376824632, −9.890191320677335, −9.539661482188799, −8.735345352121969, −8.274736606381152, −7.820258274147562, −7.449886245066787, −6.557320406461686, −6.051693667168642, −5.472997811850877, −5.270983704820010, −4.756701986742754, −3.652163794251233, −3.201466869702444, −2.686603525223945, −2.037711020293502, −1.219915419091420, −0.6300360510121311, 0.6300360510121311, 1.219915419091420, 2.037711020293502, 2.686603525223945, 3.201466869702444, 3.652163794251233, 4.756701986742754, 5.270983704820010, 5.472997811850877, 6.051693667168642, 6.557320406461686, 7.449886245066787, 7.820258274147562, 8.274736606381152, 8.735345352121969, 9.539661482188799, 9.890191320677335, 10.21270376824632, 10.84106350418483, 11.48742092078936, 11.77622181925723, 12.23508715093456, 13.05479452764172, 13.38743470588570, 13.89870766194053

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