Properties

Degree 2
Conductor $ 7 \cdot 19 $
Sign $0.895 - 0.444i$
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)4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s − 0.999·20-s + (0.5 + 0.866i)23-s + 0.999·28-s + (0.499 − 0.866i)35-s + 0.999·36-s − 43-s + (0.499 + 0.866i)44-s + (0.499 − 0.866i)45-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s − 0.999·20-s + (0.5 + 0.866i)23-s + 0.999·28-s + (0.499 − 0.866i)35-s + 0.999·36-s − 43-s + (0.499 + 0.866i)44-s + (0.499 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(133\)    =    \(7 \cdot 19\)
\( \varepsilon \)  =  $0.895 - 0.444i$
motivic weight  =  \(0\)
character  :  $\chi_{133} (18, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 133,\ (\ :0),\ 0.895 - 0.444i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5751133659\)
\(L(\frac12)\)  \(\approx\)  \(0.5751133659\)
\(L(1)\)   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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−13.51536939115785482243110465220, −12.82004215862498781437345667003, −11.46539040593524054685473501032, −10.59528197589011550738162946023, −9.311234802124883832962565317195, −8.448405537237828755980821676130, −6.94213003987707110179046690510, −6.18638201546974727841309198593, −4.06434645796077638680479019311, −3.09997349348192763214703328180, 2.16502060743839806249047133845, 4.70809796419164559349036462342, 5.42331299502316119487515539401, 6.69334998687290640538404127101, 8.586299636721084614348707081449, 9.239794672657171368592271462543, 10.10117837795661673178713172310, 11.42606310251343268145856726421, 12.61363320452011912066508697198, 13.42880854240873474245589123863

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