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Properties

Label 2-2432-8.5-c1-0-3
Degree $2$
Conductor $2432$
Sign $i$
Analytic cond. $19.4196$
Root an. cond. $4.40676$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.30i·3-s + 2.55i·5-s + 1.32·7-s − 7.91·9-s − 2.51i·11-s + 1.22i·13-s − 8.43·15-s + 0.210·17-s + i·19-s + 4.37i·21-s − 8.11·23-s − 1.51·25-s − 16.2i·27-s − 5.97i·29-s − 6.01·31-s + ⋯
L(s)  = 1  + 1.90i·3-s + 1.14i·5-s + 0.500·7-s − 2.63·9-s − 0.759i·11-s + 0.340i·13-s − 2.17·15-s + 0.0511·17-s + 0.229i·19-s + 0.955i·21-s − 1.69·23-s − 0.303·25-s − 3.12i·27-s − 1.10i·29-s − 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $i$
Analytic conductor: \(19.4196\)
Root analytic conductor: \(4.40676\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2432,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5670053817\)
\(L(\frac12)\) \(\approx\) \(0.5670053817\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - iT \)
good3 \( 1 - 3.30iT - 3T^{2} \)
5 \( 1 - 2.55iT - 5T^{2} \)
7 \( 1 - 1.32T + 7T^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 - 1.22iT - 13T^{2} \)
17 \( 1 - 0.210T + 17T^{2} \)
23 \( 1 + 8.11T + 23T^{2} \)
29 \( 1 + 5.97iT - 29T^{2} \)
31 \( 1 + 6.01T + 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 + 0.996T + 41T^{2} \)
43 \( 1 - 7.83iT - 43T^{2} \)
47 \( 1 - 0.910T + 47T^{2} \)
53 \( 1 + 3.32iT - 53T^{2} \)
59 \( 1 - 2.04iT - 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 4.50iT - 83T^{2} \)
89 \( 1 - 6.64T + 89T^{2} \)
97 \( 1 - 9.92T + 97T^{2} \)
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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

−9.658018302627593291415201091314, −8.904199657048352811622498724122, −8.190851892012580422782658753968, −7.31830638819223799531398372503, −5.92745022613581761450742028812, −5.81264110100312688509704529822, −4.52697780507453524001683273214, −3.93657571682466919174407941981, −3.20149494471153202690208114362, −2.27482797605108030577759255340, 0.17989429390841665178433837330, 1.47426112193773438982692338406, 1.88726960429090926987106521426, 3.16828396620890574849272390089, 4.59171695164788041710735843957, 5.35720794827260707897244484658, 6.10305481975676027414242594233, 7.00550830407503819199385645368, 7.60628654620479700194840238259, 8.326998552131481618002528522434

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