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Properties

Label 2-403-31.5-c1-0-0
Degree $2$
Conductor $403$
Sign $-0.891 + 0.453i$
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  − 1.54·2-s + (−0.945 + 1.63i)3-s + 0.374·4-s + (0.246 + 0.427i)5-s + (1.45 − 2.52i)6-s + (−2.41 + 4.17i)7-s + 2.50·8-s + (−0.288 − 0.500i)9-s + (−0.380 − 0.658i)10-s + (0.323 + 0.560i)11-s + (−0.353 + 0.613i)12-s + (0.5 + 0.866i)13-s + (3.71 − 6.43i)14-s − 0.933·15-s − 4.60·16-s + (−1.77 + 3.07i)17-s + ⋯
L(s)  = 1  − 1.08·2-s + (−0.546 + 0.945i)3-s + 0.187·4-s + (0.110 + 0.191i)5-s + (0.594 − 1.03i)6-s + (−0.911 + 1.57i)7-s + 0.885·8-s + (−0.0962 − 0.166i)9-s + (−0.120 − 0.208i)10-s + (0.0976 + 0.169i)11-s + (−0.102 + 0.176i)12-s + (0.138 + 0.240i)13-s + (0.993 − 1.72i)14-s − 0.241·15-s − 1.15·16-s + (−0.431 + 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0749341 - 0.312881i\)
\(L(\frac12)\) \(\approx\) \(0.0749341 - 0.312881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.48 + 0.974i)T \)
good2 \( 1 + 1.54T + 2T^{2} \)
3 \( 1 + (0.945 - 1.63i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.246 - 0.427i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.41 - 4.17i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.323 - 0.560i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.77 - 3.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.462 - 0.800i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
37 \( 1 + (-4.36 + 7.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.938 - 1.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.59 + 2.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 + (0.535 + 0.926i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.62 - 2.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 6.17T + 61T^{2} \)
67 \( 1 + (-6.54 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.81 - 13.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.94 + 5.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.56 + 6.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.37 - 11.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 + 7.58T + 97T^{2} \)
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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

−11.24782548390451709616936981512, −10.78848489259087899410538860283, −9.698069669161865171556577808706, −9.266559859346276132771617658132, −8.574208433776528516216429403598, −7.22104497189845483787516469357, −6.03024280554491976773558785127, −5.14320445855805808837776649957, −3.87164500797382812449271417801, −2.20087005733484981680993561089, 0.35345046049878403946804665988, 1.34238885266529567010874882418, 3.52538179545505029878622222726, 4.93314959170226661085528116488, 6.46534652887302031277579041254, 7.18491057390312949435174916362, 7.68079102705277014735564493300, 9.181438005615113813026054431170, 9.582739363361287078100587409084, 10.86637645044044727292678382682

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