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Properties

Label 2-403-31.2-c1-0-14
Degree $2$
Conductor $403$
Sign $0.936 - 0.349i$
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  + (0.607 + 0.441i)2-s + (0.0305 − 0.0222i)3-s + (−0.443 − 1.36i)4-s + 2.26·5-s + 0.0284·6-s + (1.52 + 4.68i)7-s + (0.797 − 2.45i)8-s + (−0.926 + 2.85i)9-s + (1.37 + 1.00i)10-s + (−0.967 − 2.97i)11-s + (−0.0439 − 0.0319i)12-s + (0.809 − 0.587i)13-s + (−1.14 + 3.52i)14-s + (0.0692 − 0.0503i)15-s + (−0.754 + 0.547i)16-s + (1.95 − 6.01i)17-s + ⋯
L(s)  = 1  + (0.429 + 0.312i)2-s + (0.0176 − 0.0128i)3-s + (−0.221 − 0.682i)4-s + 1.01·5-s + 0.0115·6-s + (0.575 + 1.77i)7-s + (0.281 − 0.867i)8-s + (−0.308 + 0.950i)9-s + (0.435 + 0.316i)10-s + (−0.291 − 0.897i)11-s + (−0.0126 − 0.00921i)12-s + (0.224 − 0.163i)13-s + (−0.305 + 0.941i)14-s + (0.0178 − 0.0129i)15-s + (−0.188 + 0.136i)16-s + (0.474 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.936 - 0.349i$
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: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95009 + 0.351829i\)
\(L(\frac12)\) \(\approx\) \(1.95009 + 0.351829i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.18 + 2.02i)T \)
good2 \( 1 + (-0.607 - 0.441i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.0305 + 0.0222i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - 2.26T + 5T^{2} \)
7 \( 1 + (-1.52 - 4.68i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.967 + 2.97i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-1.95 + 6.01i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.42 - 4.66i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.716 - 2.20i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.68 + 1.22i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 3.51T + 37T^{2} \)
41 \( 1 + (3.56 + 2.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.60 + 4.79i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.28 + 3.11i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.96 - 9.13i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.27 + 3.83i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 + (-1.27 + 3.93i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.29 - 13.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.40 + 7.39i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (13.0 + 9.45i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.889 - 2.73i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.93 + 9.03i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.43602334667957908374905070331, −10.25502394346895760846184201451, −9.517923724747440182371169496365, −8.666172218941067674523591349281, −7.62288110857332394212477269745, −5.98412647443689003113841953378, −5.51831250211520807391379396083, −5.09681991804722536521896919518, −2.96903950089556296350481870749, −1.72831962386918688826173930352, 1.50477235661408140540086395123, 3.19001325073416851374496788848, 4.19006258093417677987718661771, 5.12606843406981331462070167509, 6.54661564507813528990734782548, 7.46133413087610330357757637518, 8.393544048750157567575487564774, 9.602321229471653532811516592724, 10.30847904816302933210476508445, 11.26178368187842578407256481310

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