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Properties

Label 2-403-31.8-c1-0-18
Degree $2$
Conductor $403$
Sign $0.672 + 0.739i$
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  + (−0.599 + 1.84i)2-s + (−0.251 − 0.774i)3-s + (−1.42 − 1.03i)4-s − 3.24·5-s + 1.57·6-s + (1.80 + 1.31i)7-s + (−0.373 + 0.271i)8-s + (1.89 − 1.37i)9-s + (1.94 − 5.98i)10-s + (−4.60 − 3.34i)11-s + (−0.443 + 1.36i)12-s + (−0.309 − 0.951i)13-s + (−3.50 + 2.54i)14-s + (0.816 + 2.51i)15-s + (−1.36 − 4.20i)16-s + (0.131 − 0.0953i)17-s + ⋯
L(s)  = 1  + (−0.423 + 1.30i)2-s + (−0.145 − 0.446i)3-s + (−0.712 − 0.517i)4-s − 1.45·5-s + 0.644·6-s + (0.683 + 0.496i)7-s + (−0.132 + 0.0959i)8-s + (0.630 − 0.457i)9-s + (0.614 − 1.89i)10-s + (−1.38 − 1.00i)11-s + (−0.127 + 0.393i)12-s + (−0.0857 − 0.263i)13-s + (−0.937 + 0.681i)14-s + (0.210 + 0.648i)15-s + (−0.341 − 1.05i)16-s + (0.0318 − 0.0231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.672 + 0.739i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.672 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397561 - 0.175764i\)
\(L(\frac12)\) \(\approx\) \(0.397561 - 0.175764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-4.25 + 3.58i)T \)
good2 \( 1 + (0.599 - 1.84i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.251 + 0.774i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 3.24T + 5T^{2} \)
7 \( 1 + (-1.80 - 1.31i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (4.60 + 3.34i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (-0.131 + 0.0953i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.913 + 2.81i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.262 + 0.190i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.48 + 7.65i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 6.49T + 37T^{2} \)
41 \( 1 + (1.57 - 4.84i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.06 - 6.36i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.03 + 6.25i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.67 + 1.21i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.660 - 2.03i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + 7.78T + 67T^{2} \)
71 \( 1 + (3.62 - 2.63i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.13 - 1.54i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.86 - 3.53i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.47 - 7.63i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (13.1 + 9.55i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.98 - 4.34i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.48398932878796295902269707237, −10.10848383060066739568120606101, −8.722166769288333968255610270822, −8.026586146769269502588289680555, −7.66415101461119073570973772817, −6.64548532985469246996947857925, −5.59130225402284431073639430314, −4.55301384842637154296741483392, −2.93614633806512702159543787896, −0.33535081487329329716252307869, 1.65101621849351835022734296007, 3.20774430625029809893308300166, 4.29666263257400489913230922178, 4.97825061344858323698484429319, 7.14721237609476431875451118586, 7.78563737688430927727964206707, 8.766496170774453280286348661991, 10.11650940702486844745585937741, 10.51601236179519384591629373258, 11.14780332978964294976972238969

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