Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.445 - 0.895i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48i·2-s + 2.99·3-s − 0.218·4-s − 0.692i·5-s + 4.46i·6-s − 1.15i·7-s + 2.65i·8-s + 5.98·9-s + 1.03·10-s − 2.69i·11-s − 0.655·12-s + (−3.22 − 1.60i)13-s + 1.71·14-s − 2.07i·15-s − 4.38·16-s − 4.04·17-s + ⋯
L(s)  = 1  + 1.05i·2-s + 1.73·3-s − 0.109·4-s − 0.309i·5-s + 1.82i·6-s − 0.435i·7-s + 0.938i·8-s + 1.99·9-s + 0.326·10-s − 0.812i·11-s − 0.189·12-s + (−0.895 − 0.445i)13-s + 0.458·14-s − 0.536i·15-s − 1.09·16-s − 0.980·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.445 - 0.895i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.445 - 0.895i)$
$L(1)$  $\approx$  $2.08613 + 1.29161i$
$L(\frac12)$  $\approx$  $2.08613 + 1.29161i$
$L(\frac{3}{2})$   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 \( 1 + (3.22 + 1.60i)T \)
31 \( 1 + iT \)
good2 \( 1 - 1.48iT - 2T^{2} \)
3 \( 1 - 2.99T + 3T^{2} \)
5 \( 1 + 0.692iT - 5T^{2} \)
7 \( 1 + 1.15iT - 7T^{2} \)
11 \( 1 + 2.69iT - 11T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
19 \( 1 - 5.06iT - 19T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 + 8.56T + 29T^{2} \)
37 \( 1 + 5.88iT - 37T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + 3.63T + 43T^{2} \)
47 \( 1 - 0.848iT - 47T^{2} \)
53 \( 1 - 1.67T + 53T^{2} \)
59 \( 1 + 1.68iT - 59T^{2} \)
61 \( 1 - 2.00T + 61T^{2} \)
67 \( 1 - 8.32iT - 67T^{2} \)
71 \( 1 - 5.15iT - 71T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 0.873T + 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 - 3.90iT - 89T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−11.34153434335108888802065925478, −10.21831773154276415899926300993, −9.173096580697196698005303493206, −8.460130102530426809905090205266, −7.75211752085015854343300427172, −7.10760620793272296450003781765, −5.85449129712359117573218631581, −4.52282045609176163796053975918, −3.28054366081500727197004155129, −2.06357160758243384118352659414, 2.05450961566423557475382835359, 2.50675260774895532121658345883, 3.63888166780209979448379763620, 4.69843476103092769649233757220, 6.88154029244262590398557760793, 7.32640760987596281957566672381, 8.763369916152169763707783428973, 9.314925591688994738093391640821, 10.06619062324321540108595717345, 11.03658866416983255892282027953

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