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Properties

Label 2-403-13.12-c1-0-31
Degree $2$
Conductor $403$
Sign $-0.984 + 0.174i$
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  − 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

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.984 + 0.174i$
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: \((2,\ 403,\ (\ :1/2),\ -0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160859 - 1.82435i\)
\(L(\frac12)\) \(\approx\) \(0.160859 - 1.82435i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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