Properties

Label 2-31939-1.1-c1-0-2
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

−15.55014575348081, −14.55817991933032, −14.31944444351808, −13.58451102114650, −13.14592221952447, −12.70573050725389, −12.30489352964371, −11.40735908017762, −10.89137857400089, −10.48688962741210, −9.793930016818292, −9.503613269422172, −8.993601481517144, −8.708362808764819, −7.489171915720584, −7.008259593524127, −6.299678368647207, −6.082659321336762, −5.364591853536502, −5.001757809026971, −3.824501466370438, −3.533129843609955, −2.425201998509312, −1.436896363745034, −0.9175674879045381, 0, 0.9175674879045381, 1.436896363745034, 2.425201998509312, 3.533129843609955, 3.824501466370438, 5.001757809026971, 5.364591853536502, 6.082659321336762, 6.299678368647207, 7.008259593524127, 7.489171915720584, 8.708362808764819, 8.993601481517144, 9.503613269422172, 9.793930016818292, 10.48688962741210, 10.89137857400089, 11.40735908017762, 12.30489352964371, 12.70573050725389, 13.14592221952447, 13.58451102114650, 14.31944444351808, 14.55817991933032, 15.55014575348081

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