Properties

Degree 2
Conductor $ 7 \cdot 19 $
Sign $0.305 - 0.952i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + i·7-s − 8-s + (−0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s i·19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)23-s + (0.866 + 0.499i)24-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + i·7-s − 8-s + (−0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s i·19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)23-s + (0.866 + 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(133\)    =    \(7 \cdot 19\)
\( \varepsilon \)  =  $0.305 - 0.952i$
motivic weight  =  \(0\)
character  :  $\chi_{133} (125, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 133,\ (\ :0),\ 0.305 - 0.952i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4405601059\)
\(L(\frac12)\)  \(\approx\)  \(0.4405601059\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - iT \)
19 \( 1 + iT \)
good2 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−13.72272730390624963018605645142, −12.58533157210974718606194468804, −11.74517020196033499914094634297, −10.71499554360821088389618511129, −9.239053692940039465874360737458, −8.456438373398409426979478515721, −6.86187592916030067191584686767, −6.30950169152618461941082530168, −5.43296173482710314264454112292, −2.71044367256582579166473223063, 1.69683603271289636518353803388, 3.99891080945266620293898225975, 5.53890719816988880382030444731, 6.43424301154797598532413666283, 8.385138826224672845299114718856, 9.656034681280208466440361484716, 10.28944621613027588315661346404, 11.08243332735907686790328381905, 11.87241087780795391720983449716, 13.23459958368733222205607303095

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