Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.0999 - 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.85·2-s + (0.240 − 0.415i)3-s + 1.42·4-s + (0.854 + 1.48i)5-s + (−0.444 + 0.770i)6-s + (−0.531 + 0.921i)7-s + 1.05·8-s + (1.38 + 2.39i)9-s + (−1.58 − 2.74i)10-s + (−1.35 − 2.34i)11-s + (0.343 − 0.594i)12-s + (−0.5 − 0.866i)13-s + (0.984 − 1.70i)14-s + 0.821·15-s − 4.81·16-s + (−2.30 + 3.98i)17-s + ⋯
L(s)  = 1  − 1.30·2-s + (0.138 − 0.240i)3-s + 0.714·4-s + (0.382 + 0.662i)5-s + (−0.181 + 0.314i)6-s + (−0.201 + 0.348i)7-s + 0.374·8-s + (0.461 + 0.799i)9-s + (−0.500 − 0.867i)10-s + (−0.407 − 0.706i)11-s + (0.0990 − 0.171i)12-s + (−0.138 − 0.240i)13-s + (0.263 − 0.455i)14-s + 0.212·15-s − 1.20·16-s + (−0.558 + 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0999 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0999 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.0999 - 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.0999 - 0.994i)\)
\(L(1)\)  \(\approx\)  \(0.468633 + 0.423922i\)
\(L(\frac12)\)  \(\approx\)  \(0.468633 + 0.423922i\)
\(L(\frac{3}{2})\)   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 \( 1 + (3.77 - 4.08i)T \)
good2 \( 1 + 1.85T + 2T^{2} \)
3 \( 1 + (-0.240 + 0.415i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.854 - 1.48i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.531 - 0.921i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.35 + 2.34i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.30 - 3.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.35 - 4.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.635T + 23T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
37 \( 1 + (5.38 - 9.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.87 - 4.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.63 - 6.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.75T + 47T^{2} \)
53 \( 1 + (0.713 + 1.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.20 - 2.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + (2.39 + 4.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.06 + 10.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.91 + 6.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.09 - 5.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.24 - 7.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.00T + 89T^{2} \)
97 \( 1 + 9.12T + 97T^{2} \)
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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

−10.83942034950444764838937913483, −10.52403904774791521721561145897, −9.777691228144992927440569491613, −8.443495954338225241824518361350, −8.206342786220155916767543979783, −6.97828621008759611469413328111, −6.14035642840067394785617402529, −4.66391732899853111723704163119, −2.86645559073583732776438670594, −1.64114733283905714613143450961, 0.64270724016019112699910672773, 2.23571966893852891273220142023, 4.15263684930393412888274006949, 5.10680956941729110604160948354, 6.84217162253040557710195510353, 7.31930775338651493451342331214, 8.814483742303696767194539841719, 9.075004586977919313669547271363, 9.958060383438939459079643099595, 10.60384824913286097080184702826

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