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Properties

Label 2-403-13.10-c1-0-21
Degree $2$
Conductor $403$
Sign $0.627 + 0.778i$
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  + (0.932 − 0.538i)2-s + (−0.184 − 0.320i)3-s + (−0.420 + 0.727i)4-s − 1.90i·5-s + (−0.344 − 0.199i)6-s + (1.31 + 0.758i)7-s + 3.05i·8-s + (1.43 − 2.47i)9-s + (−1.02 − 1.77i)10-s + (2.18 − 1.26i)11-s + 0.310·12-s + (1.25 − 3.37i)13-s + 1.63·14-s + (−0.608 + 0.351i)15-s + (0.806 + 1.39i)16-s + (0.0757 − 0.131i)17-s + ⋯
L(s)  = 1  + (0.659 − 0.380i)2-s + (−0.106 − 0.184i)3-s + (−0.210 + 0.363i)4-s − 0.850i·5-s + (−0.140 − 0.0813i)6-s + (0.496 + 0.286i)7-s + 1.08i·8-s + (0.477 − 0.826i)9-s + (−0.323 − 0.560i)10-s + (0.658 − 0.379i)11-s + 0.0897·12-s + (0.348 − 0.937i)13-s + 0.436·14-s + (−0.157 + 0.0907i)15-s + (0.201 + 0.349i)16-s + (0.0183 − 0.0318i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.627 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71398 - 0.820205i\)
\(L(\frac12)\) \(\approx\) \(1.71398 - 0.820205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.25 + 3.37i)T \)
31 \( 1 + iT \)
good2 \( 1 + (-0.932 + 0.538i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.184 + 0.320i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.90iT - 5T^{2} \)
7 \( 1 + (-1.31 - 0.758i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 + 1.26i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.0757 + 0.131i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.40 - 0.810i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.187 - 0.324i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.714 + 1.23i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (7.53 - 4.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.27 + 0.736i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.42 - 7.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.07iT - 47T^{2} \)
53 \( 1 - 9.11T + 53T^{2} \)
59 \( 1 + (-9.69 - 5.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.78 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.34 - 4.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.29 + 3.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.47iT - 73T^{2} \)
79 \( 1 + 7.61T + 79T^{2} \)
83 \( 1 + 6.72iT - 83T^{2} \)
89 \( 1 + (-1.49 + 0.863i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.60 + 4.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.68942489595686059163211610712, −10.35873616956266042158056049285, −9.072654499189947487696707933017, −8.530571379970530186289744706137, −7.52035742746607668426512150858, −6.08560488120469234159640272521, −5.13324777381827519617070542538, −4.15716998602066027486748814398, −3.12326500254930095589065852430, −1.28835429484036890294527245143, 1.76187218991112669586024948600, 3.68688970033013005467939762030, 4.57129481467070961691803144776, 5.49389548899801838350558645042, 6.81951351835173861529705757784, 7.14759390756702347768515802904, 8.675325299743808528401318203660, 9.762887257794154824334906076551, 10.53645712454396539589080229191, 11.25482176415950999099632524420

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