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Properties

Label 2-403-31.4-c1-0-14
Degree $2$
Conductor $403$
Sign $0.816 - 0.577i$
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  + (0.455 + 1.40i)2-s + (0.322 − 0.991i)3-s + (−0.140 + 0.102i)4-s − 2.15·5-s + 1.53·6-s + (0.0381 − 0.0277i)7-s + (2.17 + 1.58i)8-s + (1.54 + 1.12i)9-s + (−0.982 − 3.02i)10-s + (4.86 − 3.53i)11-s + (0.0560 + 0.172i)12-s + (0.309 − 0.951i)13-s + (0.0562 + 0.0408i)14-s + (−0.694 + 2.13i)15-s + (−1.33 + 4.10i)16-s + (−0.746 − 0.542i)17-s + ⋯
L(s)  = 1  + (0.322 + 0.991i)2-s + (0.185 − 0.572i)3-s + (−0.0703 + 0.0511i)4-s − 0.964·5-s + 0.627·6-s + (0.0144 − 0.0104i)7-s + (0.770 + 0.559i)8-s + (0.516 + 0.374i)9-s + (−0.310 − 0.956i)10-s + (1.46 − 1.06i)11-s + (0.0161 + 0.0497i)12-s + (0.0857 − 0.263i)13-s + (0.0150 + 0.0109i)14-s + (−0.179 + 0.552i)15-s + (−0.333 + 1.02i)16-s + (−0.181 − 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74021 + 0.553403i\)
\(L(\frac12)\) \(\approx\) \(1.74021 + 0.553403i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.38 + 4.42i)T \)
good2 \( 1 + (-0.455 - 1.40i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.322 + 0.991i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 2.15T + 5T^{2} \)
7 \( 1 + (-0.0381 + 0.0277i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-4.86 + 3.53i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (0.746 + 0.542i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.533 - 1.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.50 - 2.54i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.63 - 5.04i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 2.43T + 37T^{2} \)
41 \( 1 + (3.39 + 10.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.30 - 4.02i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.03 - 6.27i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.98 + 3.62i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.87 + 8.85i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + (8.93 + 6.49i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.31 + 0.957i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.71 + 3.42i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.96 - 6.04i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.39 - 1.73i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.39 - 1.74i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.37831015828965102646315969006, −10.71642084381198989775578183180, −9.179543733599482510632242988825, −8.199676382817823957703927434557, −7.49243627728604031010894186403, −6.76758576302272096913460301320, −5.84844307586453961045287923946, −4.58624345530304295098711890201, −3.47405394337113041334979820045, −1.47080087626124732522935131269, 1.53099064486970250835430678672, 3.19131038493092110814645696605, 4.14062835314386668751466591169, 4.56165125435631936070881090216, 6.71126138065019459111159290967, 7.28252702068321322360291850344, 8.689214207599176731496499196541, 9.607114500977016935022360524023, 10.35349276173752325922541643589, 11.40409094109850489405048375984

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