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Properties

Label 2-403-31.8-c1-0-14
Degree $2$
Conductor $403$
Sign $0.858 - 0.513i$
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  + (−0.365 + 1.12i)2-s + (−0.709 − 2.18i)3-s + (0.485 + 0.352i)4-s + 3.83·5-s + 2.71·6-s + (3.69 + 2.68i)7-s + (−2.48 + 1.80i)8-s + (−1.83 + 1.33i)9-s + (−1.40 + 4.31i)10-s + (−2.22 − 1.61i)11-s + (0.425 − 1.31i)12-s + (−0.309 − 0.951i)13-s + (−4.37 + 3.17i)14-s + (−2.72 − 8.37i)15-s + (−0.753 − 2.31i)16-s + (−3.58 + 2.60i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.795i)2-s + (−0.409 − 1.26i)3-s + (0.242 + 0.176i)4-s + 1.71·5-s + 1.10·6-s + (1.39 + 1.01i)7-s + (−0.879 + 0.639i)8-s + (−0.611 + 0.444i)9-s + (−0.443 + 1.36i)10-s + (−0.669 − 0.486i)11-s + (0.122 − 0.378i)12-s + (−0.0857 − 0.263i)13-s + (−1.16 + 0.849i)14-s + (−0.702 − 2.16i)15-s + (−0.188 − 0.579i)16-s + (−0.869 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.858 - 0.513i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56342 + 0.431689i\)
\(L(\frac12)\) \(\approx\) \(1.56342 + 0.431689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.67 + 4.88i)T \)
good2 \( 1 + (0.365 - 1.12i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.709 + 2.18i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 - 3.83T + 5T^{2} \)
7 \( 1 + (-3.69 - 2.68i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (2.22 + 1.61i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (3.58 - 2.60i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.987 - 3.03i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.54 + 2.57i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.47 - 7.60i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 2.91T + 37T^{2} \)
41 \( 1 + (-3.88 + 11.9i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-1.32 + 4.08i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.73 + 5.35i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.60 - 1.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.68 + 8.24i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 0.710T + 61T^{2} \)
67 \( 1 + 7.83T + 67T^{2} \)
71 \( 1 + (4.48 - 3.25i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.50 - 6.17i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.126 + 0.0917i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.04 - 3.21i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.33 + 5.32i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.528 - 0.384i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.32868923860863492822774961449, −10.64796190979977957155056841415, −9.041422077755416957166392486416, −8.461080547449624338912829560045, −7.52412829096133580073425861460, −6.55220840847706452433769796477, −5.72135315838139832444533573934, −5.36005894339674920713117321004, −2.40972847315183133204653548474, −1.81012078364989902841471476688, 1.49794642631211521711760590922, 2.63612890989323648430376946584, 4.48357639784794063544996090869, 5.06865624146199981382419602853, 6.20464027815030826509744464648, 7.41339128142015167436285161415, 9.112145408849375484122830515093, 9.682166381078475135422026585303, 10.34682079615694261244689536139, 11.04973850031489924326360173544

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