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Properties

Label 2-403-31.8-c1-0-11
Degree $2$
Conductor $403$
Sign $0.702 + 0.711i$
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  + (0.774 − 2.38i)2-s + (0.754 + 2.32i)3-s + (−3.46 − 2.51i)4-s − 0.580·5-s + 6.11·6-s + (3.63 + 2.64i)7-s + (−4.63 + 3.36i)8-s + (−2.38 + 1.73i)9-s + (−0.449 + 1.38i)10-s + (2.18 + 1.58i)11-s + (3.23 − 9.94i)12-s + (−0.309 − 0.951i)13-s + (9.12 − 6.62i)14-s + (−0.437 − 1.34i)15-s + (1.79 + 5.51i)16-s + (3.15 − 2.29i)17-s + ⋯
L(s)  = 1  + (0.547 − 1.68i)2-s + (0.435 + 1.33i)3-s + (−1.73 − 1.25i)4-s − 0.259·5-s + 2.49·6-s + (1.37 + 0.999i)7-s + (−1.63 + 1.19i)8-s + (−0.796 + 0.578i)9-s + (−0.142 + 0.437i)10-s + (0.657 + 0.477i)11-s + (0.932 − 2.87i)12-s + (−0.0857 − 0.263i)13-s + (2.43 − 1.77i)14-s + (−0.112 − 0.347i)15-s + (0.447 + 1.37i)16-s + (0.764 − 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.702 + 0.711i$
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: \((2,\ 403,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90583 - 0.796258i\)
\(L(\frac12)\) \(\approx\) \(1.90583 - 0.796258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.10 + 3.76i)T \)
good2 \( 1 + (-0.774 + 2.38i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.754 - 2.32i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 0.580T + 5T^{2} \)
7 \( 1 + (-3.63 - 2.64i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-2.18 - 1.58i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (-3.15 + 2.29i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0984 - 0.303i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.78 - 1.29i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.54 + 4.76i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 9.52T + 37T^{2} \)
41 \( 1 + (0.674 - 2.07i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-0.100 + 0.310i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-1.22 - 3.76i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.45 + 2.50i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.90 + 8.93i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + (2.80 - 2.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.38 + 3.91i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (7.76 - 5.64i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.44 + 4.44i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.22 + 4.52i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (11.4 + 8.31i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.34810953049628451796713020380, −10.26301058353911326080608766582, −9.675370037933343880987217994914, −8.898578862452608705341497944641, −7.927588751311205455807424880921, −5.60755156101588892165049776597, −4.75803939273718639162095541321, −4.09237334671884032062775942502, −3.00586502408029650615412128563, −1.80589392006576250543313506131, 1.43538839378800129622326190109, 3.70079089608748294408073742439, 4.72562459876276335027921288262, 5.92692746696723495306958715187, 6.95172263544942640637885457768, 7.44144751712589619249488239101, 8.230055108588606880984509908544, 8.697394338896545523682962607239, 10.49922112303589946202081000436, 11.82851521814919082837557272098

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