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Properties

Label 2-403-31.2-c1-0-31
Degree $2$
Conductor $403$
Sign $-0.569 - 0.822i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.363i)2-s + (0.309 − 0.224i)3-s + (−0.5 − 1.53i)4-s − 2.61·5-s − 0.236·6-s + (−1.19 − 3.66i)7-s + (−0.690 + 2.12i)8-s + (−0.881 + 2.71i)9-s + (1.30 + 0.951i)10-s + (0.690 + 2.12i)11-s + (−0.5 − 0.363i)12-s + (0.809 − 0.587i)13-s + (−0.736 + 2.26i)14-s + (−0.809 + 0.587i)15-s + (−1.49 + 1.08i)16-s + (−2.07 + 6.37i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.256i)2-s + (0.178 − 0.129i)3-s + (−0.250 − 0.769i)4-s − 1.17·5-s − 0.0963·6-s + (−0.450 − 1.38i)7-s + (−0.244 + 0.751i)8-s + (−0.293 + 0.904i)9-s + (0.413 + 0.300i)10-s + (0.208 + 0.641i)11-s + (−0.144 − 0.104i)12-s + (0.224 − 0.163i)13-s + (−0.196 + 0.605i)14-s + (−0.208 + 0.151i)15-s + (−0.374 + 0.272i)16-s + (−0.502 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.569 - 0.822i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (3.23 + 4.53i)T \)
good2 \( 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.309 + 0.224i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
7 \( 1 + (1.19 + 3.66i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.690 - 2.12i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (2.07 - 6.37i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.61 - 1.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.236 + 0.726i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.35 + 3.88i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 3.14T + 37T^{2} \)
41 \( 1 + (9.78 + 7.10i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.736 + 0.534i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-3.30 + 2.40i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.73 - 11.4i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.73 - 4.16i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 1.76T + 61T^{2} \)
67 \( 1 + 8.56T + 67T^{2} \)
71 \( 1 + (-5.14 + 15.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.92 - 9.00i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.45 - 7.55i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3 + 2.17i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.54 - 7.83i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.23 + 13.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.67420862585716087397580051369, −9.991695366956481218657153831344, −8.799154749548704979384082576144, −7.88174866995798798603224743548, −7.17494018187716166597745767980, −5.87628588918034514358607270707, −4.48208633667020097721719139888, −3.70353449986421535336322271474, −1.77333867030867372193539987598, 0, 3.04731355933930222633106371832, 3.56877360047291224479796960678, 5.03956062957140599161004051639, 6.43932413523730878030676757009, 7.30251873596079134810413331323, 8.421247875333838184784577305652, 8.978161753887288485540598489528, 9.516981854896133843383073812445, 11.38838737127422921342888041982

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