Properties

Label 2-31939-1.1-c1-0-4
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-s + 2·3-s − 4-s + 3·5-s − 2·6-s + 4·7-s + 3·8-s + 9-s − 3·10-s − 4·11-s − 2·12-s − 2·13-s − 4·14-s + 6·15-s − 16-s − 3·17-s − 18-s + 19-s − 3·20-s + 8·21-s + 4·22-s + 7·23-s + 6·24-s + 4·25-s + 2·26-s − 4·27-s − 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s + 1.34·5-s − 0.816·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.948·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 1.54·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s − 0.670·20-s + 1.74·21-s + 0.852·22-s + 1.45·23-s + 1.22·24-s + 4/5·25-s + 0.392·26-s − 0.769·27-s − 0.755·28-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: \(1\)
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 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−15.01059877269697, −14.69842576144412, −14.31797211854825, −13.70175671018537, −13.35622443062187, −13.07856829739937, −12.27261257841658, −11.33465651126155, −10.81335008084364, −10.43445922657055, −9.761708767706927, −9.307602977712748, −8.848880599033317, −8.333782731288707, −8.000076605589945, −7.316415523592087, −6.846879764836664, −5.519063264026636, −5.360023189903999, −4.726282611499061, −4.101916375998842, −2.915192671732627, −2.539758292997555, −1.740751495393528, −1.358821273105602, 0, 1.358821273105602, 1.740751495393528, 2.539758292997555, 2.915192671732627, 4.101916375998842, 4.726282611499061, 5.360023189903999, 5.519063264026636, 6.846879764836664, 7.316415523592087, 8.000076605589945, 8.333782731288707, 8.848880599033317, 9.307602977712748, 9.761708767706927, 10.43445922657055, 10.81335008084364, 11.33465651126155, 12.27261257841658, 13.07856829739937, 13.35622443062187, 13.70175671018537, 14.31797211854825, 14.69842576144412, 15.01059877269697

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