Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.115 + 0.993i$
Motivic weight 1
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

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.115 + 0.993i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.115 + 0.993i)\)
\(L(1)\)  \(\approx\)  \(0.433665 - 0.486882i\)
\(L(\frac12)\)  \(\approx\)  \(0.433665 - 0.486882i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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