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Properties

Label 2-403-13.12-c1-0-26
Degree $2$
Conductor $403$
Sign $-0.989 + 0.142i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  − 1.88i·2-s − 0.327·3-s − 1.57·4-s + 1.75i·5-s + 0.619i·6-s − 0.652i·7-s − 0.809i·8-s − 2.89·9-s + 3.32·10-s − 4.04i·11-s + 0.515·12-s + (−0.515 − 3.56i)13-s − 1.23·14-s − 0.576i·15-s − 4.67·16-s − 2.05·17-s + ⋯
L(s)  = 1  − 1.33i·2-s − 0.189·3-s − 0.785·4-s + 0.786i·5-s + 0.253i·6-s − 0.246i·7-s − 0.286i·8-s − 0.964·9-s + 1.05·10-s − 1.21i·11-s + 0.148·12-s + (−0.142 − 0.989i)13-s − 0.329·14-s − 0.148i·15-s − 1.16·16-s − 0.497·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0703757 - 0.979748i\)
\(L(\frac12)\) \(\approx\) \(0.0703757 - 0.979748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.515 + 3.56i)T \)
31 \( 1 - iT \)
good2 \( 1 + 1.88iT - 2T^{2} \)
3 \( 1 + 0.327T + 3T^{2} \)
5 \( 1 - 1.75iT - 5T^{2} \)
7 \( 1 + 0.652iT - 7T^{2} \)
11 \( 1 + 4.04iT - 11T^{2} \)
17 \( 1 + 2.05T + 17T^{2} \)
19 \( 1 + 7.52iT - 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
37 \( 1 - 4.69iT - 37T^{2} \)
41 \( 1 - 5.09iT - 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 - 1.20iT - 47T^{2} \)
53 \( 1 - 0.419T + 53T^{2} \)
59 \( 1 - 10.2iT - 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 11.5iT - 67T^{2} \)
71 \( 1 - 6.53iT - 71T^{2} \)
73 \( 1 + 0.170iT - 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 - 7.61iT - 89T^{2} \)
97 \( 1 + 18.9iT - 97T^{2} \)
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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

−10.85901605172508877856681462374, −10.46465198526361978530628784333, −9.204898747076564123965513070669, −8.343193244330552446004974132929, −6.95550839231033821213511419528, −6.03578955171200017613512431398, −4.66265505874162637322310970270, −3.07131329068221567178302401449, −2.80664087413154612798097482387, −0.63914791125131542572731536242, 2.13311334932139062868304321626, 4.26964258794059452784282415867, 5.16905937772329673921587002881, 6.04301779352242820796867077375, 6.93466466485604739681420106416, 7.970212756469438463448574177342, 8.724657473575092635178637817426, 9.478215499207700951142849505199, 10.76887675022927384351060012140, 12.01332353638631981974522950177

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