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Properties

Label 2-403-31.25-c1-0-4
Degree $2$
Conductor $403$
Sign $-0.115 - 0.993i$
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  − 2.00·2-s + (0.235 + 0.407i)3-s + 2.02·4-s + (−0.859 + 1.48i)5-s + (−0.472 − 0.818i)6-s + (1.82 + 3.15i)7-s − 0.0581·8-s + (1.38 − 2.40i)9-s + (1.72 − 2.98i)10-s + (0.222 − 0.386i)11-s + (0.477 + 0.827i)12-s + (−0.5 + 0.866i)13-s + (−3.65 − 6.33i)14-s − 0.809·15-s − 3.94·16-s + (0.176 + 0.305i)17-s + ⋯
L(s)  = 1  − 1.41·2-s + (0.135 + 0.235i)3-s + 1.01·4-s + (−0.384 + 0.665i)5-s + (−0.192 − 0.334i)6-s + (0.688 + 1.19i)7-s − 0.0205·8-s + (0.463 − 0.802i)9-s + (0.545 − 0.945i)10-s + (0.0671 − 0.116i)11-s + (0.137 + 0.238i)12-s + (−0.138 + 0.240i)13-s + (−0.977 − 1.69i)14-s − 0.209·15-s − 0.985·16-s + (0.0427 + 0.0741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.115 - 0.993i$
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: \((2,\ 403,\ (\ :1/2),\ -0.115 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433665 + 0.486882i\)
\(L(\frac12)\) \(\approx\) \(0.433665 + 0.486882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.84 - 4.02i)T \)
good2 \( 1 + 2.00T + 2T^{2} \)
3 \( 1 + (-0.235 - 0.407i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.859 - 1.48i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.82 - 3.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.222 + 0.386i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.176 - 0.305i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.901 - 1.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.216T + 23T^{2} \)
29 \( 1 + 8.89T + 29T^{2} \)
37 \( 1 + (-2.74 - 4.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.304 + 0.527i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.29 - 7.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + (3.42 - 5.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.69 - 6.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 + (-0.121 + 0.209i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.15 - 5.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.0149 + 0.0259i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.18 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.97 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 - 3.84T + 97T^{2} \)
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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

−11.36378401495790721162500561840, −10.41769248844881862124677228338, −9.482138314767819298221546925675, −8.918449702036880054685272748401, −8.000197837251279521917931330186, −7.17940108390538163245790365498, −6.09006756042042573381775965766, −4.60984409173294999812716968232, −3.09936993074531124011423649656, −1.60362165562047204490384751677, 0.72442242168256544961592726379, 1.96074896707424214689308616883, 4.13587708018066970548271433957, 5.01705236986126577179691172276, 6.89796340105774650527510666746, 7.74786760992071459694326112738, 8.034860833034248562923300714678, 9.153751069012531988917813167420, 10.04192310586263033497636804098, 10.83442798052071261262105189907

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