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Properties

Label 2-403-31.16-c1-0-26
Degree $2$
Conductor $403$
Sign $0.340 + 0.940i$
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  + (1.73 − 1.26i)2-s + (0.149 + 0.108i)3-s + (0.805 − 2.47i)4-s + 1.51·5-s + 0.395·6-s + (−0.190 + 0.586i)7-s + (−0.401 − 1.23i)8-s + (−0.916 − 2.82i)9-s + (2.63 − 1.91i)10-s + (0.332 − 1.02i)11-s + (0.388 − 0.282i)12-s + (0.809 + 0.587i)13-s + (0.408 + 1.25i)14-s + (0.226 + 0.164i)15-s + (1.96 + 1.42i)16-s + (−0.629 − 1.93i)17-s + ⋯
L(s)  = 1  + (1.22 − 0.891i)2-s + (0.0861 + 0.0626i)3-s + (0.402 − 1.23i)4-s + 0.678·5-s + 0.161·6-s + (−0.0720 + 0.221i)7-s + (−0.141 − 0.436i)8-s + (−0.305 − 0.940i)9-s + (0.832 − 0.604i)10-s + (0.100 − 0.308i)11-s + (0.112 − 0.0815i)12-s + (0.224 + 0.163i)13-s + (0.109 + 0.336i)14-s + (0.0584 + 0.0424i)15-s + (0.490 + 0.356i)16-s + (−0.152 − 0.469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30734 - 1.61819i\)
\(L(\frac12)\) \(\approx\) \(2.30734 - 1.61819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 - 5.26i)T \)
good2 \( 1 + (-1.73 + 1.26i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.149 - 0.108i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - 1.51T + 5T^{2} \)
7 \( 1 + (0.190 - 0.586i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.332 + 1.02i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (0.629 + 1.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.0620 - 0.0450i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.27 - 3.93i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.95 - 4.32i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + (0.255 - 0.185i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-4.97 + 3.61i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (9.59 + 6.97i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.16 - 9.73i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.969 - 0.704i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 4.08T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 + (1.73 + 5.34i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.10 - 6.48i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.511 - 1.57i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.0196 - 0.0142i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.0665 + 0.204i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.33 + 7.18i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.34555689782029409170734450174, −10.44446992169470212537994583649, −9.429397392385994684443679186414, −8.638695789687494109236533749542, −6.99526019908833060442039346559, −5.87510000874386827699991663104, −5.22880900434750777763493237680, −3.84043731327039960539460951579, −3.03807501525098213038870805704, −1.68301134351007190283808133764, 2.19044096703024415508234451688, 3.70787025169104826487922488205, 4.80212108669556263543460660983, 5.67620043426252517077085794286, 6.46114667785420087145052995000, 7.47986587672115388270664380985, 8.339346032197615252226885635103, 9.671052376703567183784068919924, 10.56308271664219983199882220689, 11.64535606989030199306771654821

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