Properties

Label 2-403-31.4-c1-0-16
Degree $2$
Conductor $403$
Sign $-0.982 - 0.186i$
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  + (−0.831 − 2.55i)2-s + (−0.796 + 2.45i)3-s + (−4.23 + 3.07i)4-s − 0.866·5-s + 6.93·6-s + (1.08 − 0.790i)7-s + (7.04 + 5.11i)8-s + (−2.95 − 2.14i)9-s + (0.720 + 2.21i)10-s + (−0.806 + 0.585i)11-s + (−4.17 − 12.8i)12-s + (0.309 − 0.951i)13-s + (−2.92 − 2.12i)14-s + (0.690 − 2.12i)15-s + (3.99 − 12.3i)16-s + (−4.93 − 3.58i)17-s + ⋯
L(s)  = 1  + (−0.587 − 1.80i)2-s + (−0.460 + 1.41i)3-s + (−2.11 + 1.53i)4-s − 0.387·5-s + 2.83·6-s + (0.411 − 0.298i)7-s + (2.48 + 1.80i)8-s + (−0.984 − 0.715i)9-s + (0.227 + 0.700i)10-s + (−0.243 + 0.176i)11-s + (−1.20 − 3.70i)12-s + (0.0857 − 0.263i)13-s + (−0.782 − 0.568i)14-s + (0.178 − 0.548i)15-s + (0.999 − 3.07i)16-s + (−1.19 − 0.870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.982 - 0.186i$
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: \((2,\ 403,\ (\ :1/2),\ -0.982 - 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0210627 + 0.223454i\)
\(L(\frac12)\) \(\approx\) \(0.0210627 + 0.223454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.833 - 5.50i)T \)
good2 \( 1 + (0.831 + 2.55i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.796 - 2.45i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 0.866T + 5T^{2} \)
7 \( 1 + (-1.08 + 0.790i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.806 - 0.585i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (4.93 + 3.58i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.793 + 2.44i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.357 - 0.259i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.71 + 8.35i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 2.91T + 37T^{2} \)
41 \( 1 + (3.33 + 10.2i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.49 + 7.67i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.69 + 8.28i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.09 - 4.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0998 + 0.307i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 4.77T + 61T^{2} \)
67 \( 1 + 3.10T + 67T^{2} \)
71 \( 1 + (8.58 + 6.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.25 - 6.72i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.44 - 3.95i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.611 - 1.88i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.96 + 2.15i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.50 + 3.27i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.67578464362875761066642198758, −10.22872571321199448798112310154, −9.283153674435999274448051547327, −8.658670736067225546266562630333, −7.43215607937217553530944031058, −5.25699303969386909020115021449, −4.37272598561157431729894608861, −3.73370763733186788018386624296, −2.34685022594791540878287326338, −0.19375479944352983077391930826, 1.58113890875336584322423422674, 4.37565609350053979025133760509, 5.62912449611362437070610439009, 6.31602854538465289766847533970, 7.06226661358794183222354800431, 7.964233812160437036328117864133, 8.379207067535912842192381970025, 9.475576789435548390182879640362, 10.82081623931401429773756653801, 11.78200345250057929853432042689

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