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Properties

Label 2-31939-1.1-c1-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s + 3·5-s + 7-s + 9-s − 3·11-s − 4·12-s + 4·13-s + 6·15-s + 4·16-s + 3·17-s − 19-s − 6·20-s + 2·21-s + 4·25-s − 4·27-s − 2·28-s − 6·29-s − 4·31-s − 6·33-s + 3·35-s − 2·36-s + 2·37-s + 8·39-s − 43-s + 6·44-s + 3·45-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.15·12-s + 1.10·13-s + 1.54·15-s + 16-s + 0.727·17-s − 0.229·19-s − 1.34·20-s + 0.436·21-s + 4/5·25-s − 0.769·27-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 1.04·33-s + 0.507·35-s − 1/3·36-s + 0.328·37-s + 1.28·39-s − 0.152·43-s + 0.904·44-s + 0.447·45-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31939,\ (\ :1/2),\ -1)\)

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)$
bad19 \( 1 + T \)
41 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.07257660287803, −14.61048063968248, −14.12850358030768, −13.82858682028287, −13.33696096153065, −12.90705710788637, −12.60485518121960, −11.49116892091406, −10.92962073539208, −10.29860447606755, −9.828581986868246, −9.237578045861150, −9.008203837899821, −8.296499449630110, −7.923460562946246, −7.395323276858597, −6.300926247989272, −5.751163261409826, −5.384023834925156, −4.661136704453545, −3.856152522085143, −3.284032355587950, −2.671836387301152, −1.795546528718987, −1.354860743881977, 0, 1.354860743881977, 1.795546528718987, 2.671836387301152, 3.284032355587950, 3.856152522085143, 4.661136704453545, 5.384023834925156, 5.751163261409826, 6.300926247989272, 7.395323276858597, 7.923460562946246, 8.296499449630110, 9.008203837899821, 9.237578045861150, 9.828581986868246, 10.29860447606755, 10.92962073539208, 11.49116892091406, 12.60485518121960, 12.90705710788637, 13.33696096153065, 13.82858682028287, 14.12850358030768, 14.61048063968248, 15.07257660287803

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