Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.989 - 0.142i$
Motivic weight 1
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

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.989 - 0.142i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.989 - 0.142i)$
$L(1)$  $\approx$  $0.0703757 + 0.979748i$
$L(\frac12)$  $\approx$  $0.0703757 + 0.979748i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;31\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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