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Properties

Label 2-403-31.25-c1-0-6
Degree $2$
Conductor $403$
Sign $0.969 + 0.245i$
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  − 1.20·2-s + (−1.24 − 2.15i)3-s − 0.547·4-s + (0.667 − 1.15i)5-s + (1.50 + 2.59i)6-s + (1.94 + 3.36i)7-s + 3.07·8-s + (−1.60 + 2.77i)9-s + (−0.804 + 1.39i)10-s + (−2.37 + 4.11i)11-s + (0.681 + 1.18i)12-s + (0.5 − 0.866i)13-s + (−2.34 − 4.05i)14-s − 3.32·15-s − 2.60·16-s + (2.91 + 5.05i)17-s + ⋯
L(s)  = 1  − 0.852·2-s + (−0.718 − 1.24i)3-s − 0.273·4-s + (0.298 − 0.516i)5-s + (0.612 + 1.06i)6-s + (0.734 + 1.27i)7-s + 1.08·8-s + (−0.533 + 0.924i)9-s + (−0.254 + 0.440i)10-s + (−0.715 + 1.23i)11-s + (0.196 + 0.340i)12-s + (0.138 − 0.240i)13-s + (−0.626 − 1.08i)14-s − 0.857·15-s − 0.651·16-s + (0.708 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.660422 - 0.0821717i\)
\(L(\frac12)\) \(\approx\) \(0.660422 - 0.0821717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-5.13 + 2.16i)T \)
good2 \( 1 + 1.20T + 2T^{2} \)
3 \( 1 + (1.24 + 2.15i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.667 + 1.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.94 - 3.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.37 - 4.11i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.91 - 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.16 + 3.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.18T + 23T^{2} \)
29 \( 1 - 4.46T + 29T^{2} \)
37 \( 1 + (2.02 + 3.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.744 - 1.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + (-1.30 + 2.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.07 - 12.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.94T + 61T^{2} \)
67 \( 1 + (-2.95 + 5.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.68 - 4.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.21 + 12.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.96 - 5.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.47 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 6.62T + 97T^{2} \)
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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

−11.23590724283952276038509866189, −10.29860318262720268197766629242, −9.200184201172269165785493285890, −8.407295699031298812947559975491, −7.71671330912339658534956149866, −6.67534877076815526603773346224, −5.41050127466317831784071479519, −4.83342707837607087198099693482, −2.20947453767380637945530106080, −1.18026291418277686579373951181, 0.814440511479119700525556088772, 3.33793518411463022614368150930, 4.60222569748655740122311693229, 5.19961183377777687190786316359, 6.65773394634693843283568574770, 7.83880941780476501756135808042, 8.641737113050868135775712690344, 9.875008709638722816173410756362, 10.32233919259970194200605034547, 10.89174721683921317152158192598

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