Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.0605 - 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.72·2-s + (−1.49 + 2.58i)3-s + 0.970·4-s + (−1.96 − 3.40i)5-s + (2.57 − 4.45i)6-s + (0.142 − 0.246i)7-s + 1.77·8-s + (−2.94 − 5.10i)9-s + (3.39 + 5.87i)10-s + (2.19 + 3.80i)11-s + (−1.44 + 2.50i)12-s + (−0.5 − 0.866i)13-s + (−0.245 + 0.425i)14-s + 11.7·15-s − 4.99·16-s + (2.72 − 4.71i)17-s + ⋯
L(s)  = 1  − 1.21·2-s + (−0.861 + 1.49i)3-s + 0.485·4-s + (−0.879 − 1.52i)5-s + (1.04 − 1.81i)6-s + (0.0538 − 0.0932i)7-s + 0.627·8-s + (−0.982 − 1.70i)9-s + (1.07 + 1.85i)10-s + (0.661 + 1.14i)11-s + (−0.417 + 0.723i)12-s + (−0.138 − 0.240i)13-s + (−0.0656 + 0.113i)14-s + 3.03·15-s − 1.24·16-s + (0.660 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.0605 - 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.0605 - 0.998i)\)
\(L(1)\)  \(\approx\)  \(0.227899 + 0.242136i\)
\(L(\frac12)\)  \(\approx\)  \(0.227899 + 0.242136i\)
\(L(\frac{3}{2})\)   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 \( 1 + (3.07 - 4.64i)T \)
good2 \( 1 + 1.72T + 2T^{2} \)
3 \( 1 + (1.49 - 2.58i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.96 + 3.40i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.142 + 0.246i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.72 + 4.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.983 - 1.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.503T + 23T^{2} \)
29 \( 1 + 1.05T + 29T^{2} \)
37 \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.42 - 4.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.52 - 4.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 + (5.72 + 9.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + (-0.707 - 1.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 9.00i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.53 - 7.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.01 - 6.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.505 + 0.875i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + 1.31T + 97T^{2} \)
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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

−11.34809450421054576004113408957, −10.29277052870744008897304493469, −9.560221613320910172652435826989, −9.111279875851753019297291618678, −8.153898157815809901766612119060, −7.12038857992286659062651139945, −5.32156868316412188115481200722, −4.66981466940181045355322697537, −3.93276846963556788523409533059, −0.995734086912398261125855202002, 0.49244628724699744124650724109, 2.08321532946690865333465629388, 3.75429684140737612615178620247, 5.83622508124815262250805457347, 6.65562515507233323837692476286, 7.40166515232848219619873810881, 7.969616733985604297350705631721, 8.964257219508848677872328112674, 10.54127945148458337950118293566, 10.94606218206254508697274951809

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