Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.474 + 0.880i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.51·2-s + (1.68 − 2.92i)3-s + 0.290·4-s + (1.24 + 2.15i)5-s + (−2.55 + 4.41i)6-s + (−1.03 + 1.79i)7-s + 2.58·8-s + (−4.18 − 7.24i)9-s + (−1.88 − 3.25i)10-s + (−2.30 − 3.99i)11-s + (0.490 − 0.848i)12-s + (−0.5 − 0.866i)13-s + (1.56 − 2.71i)14-s + 8.37·15-s − 4.49·16-s + (1.97 − 3.42i)17-s + ⋯
L(s)  = 1  − 1.07·2-s + (0.973 − 1.68i)3-s + 0.145·4-s + (0.555 + 0.962i)5-s + (−1.04 + 1.80i)6-s + (−0.391 + 0.677i)7-s + 0.914·8-s + (−1.39 − 2.41i)9-s + (−0.594 − 1.02i)10-s + (−0.695 − 1.20i)11-s + (0.141 − 0.245i)12-s + (−0.138 − 0.240i)13-s + (0.418 − 0.725i)14-s + 2.16·15-s − 1.12·16-s + (0.479 − 0.830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.474 + 0.880i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.474 + 0.880i)\)
\(L(1)\)  \(\approx\)  \(0.478457 - 0.801011i\)
\(L(\frac12)\)  \(\approx\)  \(0.478457 - 0.801011i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-5.05 + 2.32i)T \)
good2 \( 1 + 1.51T + 2T^{2} \)
3 \( 1 + (-1.68 + 2.92i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.24 - 2.15i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.03 - 1.79i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.30 + 3.99i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.97 + 3.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.35 + 5.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.227T + 23T^{2} \)
29 \( 1 + 4.45T + 29T^{2} \)
37 \( 1 + (-0.645 + 1.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.59 - 7.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.49 + 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.11T + 47T^{2} \)
53 \( 1 + (-3.23 - 5.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.62 - 2.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.21T + 61T^{2} \)
67 \( 1 + (-0.269 - 0.467i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.67 - 8.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.94 - 3.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.79 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.02 - 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 3.72T + 97T^{2} \)
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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

−10.90617355524201716629541534174, −9.598671891903427665670070571894, −9.092935706669659425130579568502, −8.123431467601825854227504419514, −7.49110343432954587274586494358, −6.63725310046016630178752309902, −5.66567950753792870125061796938, −2.98799761556776201375201268233, −2.50966494892739141745666649156, −0.794300006285590677471971039458, 1.89419886146785648365660731701, 3.67709138024520013120002419103, 4.58998794965020648539244676998, 5.41065350312480397938786552162, 7.54728981904017982917233008341, 8.206942311735540116001394032886, 9.091514161686492286963269222625, 9.876797002252673862214310461807, 9.988285498162342888923219251033, 10.83822055784619926739669967545

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