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Properties

Label 2-403-31.4-c1-0-24
Degree $2$
Conductor $403$
Sign $-0.965 - 0.258i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.53i)2-s + (−0.809 + 2.48i)3-s + (−0.5 + 0.363i)4-s − 0.381·5-s + 4.23·6-s + (−2.30 + 1.67i)7-s + (−1.80 − 1.31i)8-s + (−3.11 − 2.26i)9-s + (0.190 + 0.587i)10-s + (1.80 − 1.31i)11-s + (−0.500 − 1.53i)12-s + (−0.309 + 0.951i)13-s + (3.73 + 2.71i)14-s + (0.309 − 0.951i)15-s + (−1.50 + 4.61i)16-s + (−5.42 − 3.94i)17-s + ⋯
L(s)  = 1  + (−0.353 − 1.08i)2-s + (−0.467 + 1.43i)3-s + (−0.250 + 0.181i)4-s − 0.170·5-s + 1.72·6-s + (−0.872 + 0.634i)7-s + (−0.639 − 0.464i)8-s + (−1.03 − 0.755i)9-s + (0.0603 + 0.185i)10-s + (0.545 − 0.396i)11-s + (−0.144 − 0.444i)12-s + (−0.0857 + 0.263i)13-s + (0.998 + 0.725i)14-s + (0.0797 − 0.245i)15-s + (−0.375 + 1.15i)16-s + (−1.31 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.965 - 0.258i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.965 - 0.258i)\)

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)$
bad13 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-1.23 + 5.42i)T \)
good2 \( 1 + (0.5 + 1.53i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.809 - 2.48i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
7 \( 1 + (2.30 - 1.67i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.80 + 1.31i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (5.42 + 3.94i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.618 + 1.90i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.23 + 3.07i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.35 - 4.16i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 9.85T + 37T^{2} \)
41 \( 1 + (-0.281 - 0.865i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.73 - 11.4i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.19 + 6.74i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.736 - 0.534i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.26 - 3.88i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 6.23T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (-11.8 - 8.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.427 - 0.310i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (8.04 + 5.84i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3 + 9.23i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.04 - 2.21i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.236 + 0.171i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.79249086746883122322300283875, −9.891401282558699514009874735020, −9.345537737779184092797890606110, −8.714251004821748918159059418163, −6.71235103720606587272187125074, −5.86858731207284525319102147790, −4.51502069935152923555279608600, −3.58290050572778133579737609991, −2.45944138474087666874750680257, 0, 2.00025551590611070632357829877, 3.84294579026050579128753428219, 5.67051064014435167980319739509, 6.49976727006276343069809015171, 6.92527797167722836896630539415, 7.76975535155222827325160735093, 8.561359501563538684735142171081, 9.763532508281920351008914961839, 10.94798747012310419596198981511

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