Properties

Label 2-31939-1.1-c1-0-0
Degree $2$
Conductor $31939$
Sign $1$
Analytic cond. $255.034$
Root an. cond. $15.9697$
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  − 3-s − 2·4-s + 2·5-s + 2·7-s − 2·9-s − 2·11-s + 2·12-s + 5·13-s − 2·15-s + 4·16-s + 4·17-s + 19-s − 4·20-s − 2·21-s + 23-s − 25-s + 5·27-s − 4·28-s − 6·29-s + 8·31-s + 2·33-s + 4·35-s + 4·36-s + 8·37-s − 5·39-s − 43-s + 4·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.894·5-s + 0.755·7-s − 2/3·9-s − 0.603·11-s + 0.577·12-s + 1.38·13-s − 0.516·15-s + 16-s + 0.970·17-s + 0.229·19-s − 0.894·20-s − 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.962·27-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.348·33-s + 0.676·35-s + 2/3·36-s + 1.31·37-s − 0.800·39-s − 0.152·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31939\)    =    \(19 \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(255.034\)
Root analytic conductor: \(15.9697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31939} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 31939,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033676518\)
\(L(\frac12)\) \(\approx\) \(2.033676518\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 - T \)
41 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−14.87985453307271, −14.43982963240872, −13.86859785853340, −13.60859434599685, −13.01677733118593, −12.57473181785615, −11.69416117345087, −11.42528749762038, −10.72174647526401, −10.31139590342123, −9.588500652573828, −9.282595416046464, −8.416399972674751, −8.137900844938288, −7.646341354240347, −6.490650472413305, −6.037668632449196, −5.516747327306734, −5.150895072487945, −4.520265684132887, −3.678862958116330, −3.072584808936885, −2.133574623543278, −1.234600284353340, −0.6485131708301260, 0.6485131708301260, 1.234600284353340, 2.133574623543278, 3.072584808936885, 3.678862958116330, 4.520265684132887, 5.150895072487945, 5.516747327306734, 6.037668632449196, 6.490650472413305, 7.646341354240347, 8.137900844938288, 8.416399972674751, 9.282595416046464, 9.588500652573828, 10.31139590342123, 10.72174647526401, 11.42528749762038, 11.69416117345087, 12.57473181785615, 13.01677733118593, 13.60859434599685, 13.86859785853340, 14.43982963240872, 14.87985453307271

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