Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.604 + 0.796i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.837·2-s + (0.883 + 1.53i)3-s − 1.29·4-s + (−2.21 + 3.83i)5-s + (−0.739 − 1.28i)6-s + (−1.99 − 3.45i)7-s + 2.76·8-s + (−0.0613 + 0.106i)9-s + (1.85 − 3.20i)10-s + (−1.39 + 2.41i)11-s + (−1.14 − 1.98i)12-s + (−0.5 + 0.866i)13-s + (1.67 + 2.89i)14-s − 7.82·15-s + 0.284·16-s + (−0.576 − 0.999i)17-s + ⋯
L(s)  = 1  − 0.592·2-s + (0.510 + 0.883i)3-s − 0.649·4-s + (−0.989 + 1.71i)5-s + (−0.302 − 0.523i)6-s + (−0.754 − 1.30i)7-s + 0.976·8-s + (−0.0204 + 0.0354i)9-s + (0.586 − 1.01i)10-s + (−0.420 + 0.728i)11-s + (−0.331 − 0.573i)12-s + (−0.138 + 0.240i)13-s + (0.446 + 0.774i)14-s − 2.01·15-s + 0.0711·16-s + (−0.139 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.604 + 0.796i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.604 + 0.796i)\)
\(L(1)\)  \(\approx\)  \(0.0469677 - 0.0946438i\)
\(L(\frac12)\)  \(\approx\)  \(0.0469677 - 0.0946438i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.0190 + 5.56i)T \)
good2 \( 1 + 0.837T + 2T^{2} \)
3 \( 1 + (-0.883 - 1.53i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.21 - 3.83i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.99 + 3.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.39 - 2.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.576 + 0.999i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.68 + 2.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.745T + 23T^{2} \)
29 \( 1 + 5.02T + 29T^{2} \)
37 \( 1 + (1.55 + 2.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.81 - 8.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.51 - 7.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.11T + 47T^{2} \)
53 \( 1 + (-2.71 + 4.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.500 - 0.867i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + (0.828 - 1.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.716 - 1.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.94 - 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.40 - 2.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.39 - 9.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.37T + 89T^{2} \)
97 \( 1 + 4.35T + 97T^{2} \)
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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

−11.32694401861839105452612736741, −10.64732869787995330231000563281, −9.934651143706204250875746210735, −9.514701079449855951488512116133, −8.102736600141389554155401526540, −7.30689795532852639322287482664, −6.68586674007056924561383792033, −4.46048829279121889253105076645, −3.92952348354158701494973549715, −2.95460673664764475924854452665, 0.081023655658243178625881327101, 1.65580562950103542528505491487, 3.47473609010243750676964656858, 4.84925570310431022172848088530, 5.72325297059146479158290284802, 7.42358342289024683916485947431, 8.227116408028233318330423116162, 8.721382004804974623884538404615, 9.195641514158073862689078859867, 10.52054363548573304041563406340

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