Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.984 - 0.174i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44i·2-s + 2.35·3-s − 3.98·4-s + 2.67i·5-s + 5.76i·6-s − 0.955i·7-s − 4.86i·8-s + 2.54·9-s − 6.54·10-s − 0.164i·11-s − 9.39·12-s + (−0.630 + 3.54i)13-s + 2.33·14-s + 6.30i·15-s + 3.92·16-s + 0.846·17-s + ⋯
L(s)  = 1  + 1.73i·2-s + 1.35·3-s − 1.99·4-s + 1.19i·5-s + 2.35i·6-s − 0.361i·7-s − 1.71i·8-s + 0.849·9-s − 2.07·10-s − 0.0496i·11-s − 2.71·12-s + (−0.174 + 0.984i)13-s + 0.624·14-s + 1.62i·15-s + 0.981·16-s + 0.205·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.984 - 0.174i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.984 - 0.174i)$
$L(1)$  $\approx$  $0.160859 + 1.82435i$
$L(\frac12)$  $\approx$  $0.160859 + 1.82435i$
$L(\frac{3}{2})$   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 \( 1 + (0.630 - 3.54i)T \)
31 \( 1 + iT \)
good2 \( 1 - 2.44iT - 2T^{2} \)
3 \( 1 - 2.35T + 3T^{2} \)
5 \( 1 - 2.67iT - 5T^{2} \)
7 \( 1 + 0.955iT - 7T^{2} \)
11 \( 1 + 0.164iT - 11T^{2} \)
17 \( 1 - 0.846T + 17T^{2} \)
19 \( 1 + 4.61iT - 19T^{2} \)
23 \( 1 - 3.55T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
37 \( 1 - 7.21iT - 37T^{2} \)
41 \( 1 - 0.599iT - 41T^{2} \)
43 \( 1 + 4.14T + 43T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 - 9.33T + 53T^{2} \)
59 \( 1 + 7.75iT - 59T^{2} \)
61 \( 1 + 6.21T + 61T^{2} \)
67 \( 1 + 0.784iT - 67T^{2} \)
71 \( 1 - 2.65iT - 71T^{2} \)
73 \( 1 + 0.941iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 7.61iT - 83T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 - 6.09iT - 97T^{2} \)
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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

−11.66809484265741588479099543449, −10.41960432949321972756373837425, −9.393422386226368564949161264943, −8.699240724476778217774506136850, −7.83924025097122389818681190772, −6.93982073828828367735273356353, −6.61412484612351510929449053266, −5.01382050854399457169100119515, −3.81735194382274695705318162584, −2.62492678153008364105079806605, 1.17199129180240821136556035084, 2.45319932288708787448398262960, 3.36541940358752096007784171339, 4.41343478956782878668066935714, 5.51266617650799244755733362643, 7.71944646015373227751747143995, 8.599741941261826852751391610086, 9.052127651673320211011904214334, 9.892514866999519518058106902527, 10.71074487563909311088769929266

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