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  =  \((2,\ 133,\ (\ :0),\ 0.305 - 0.952i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5562770662\)
\(L(\frac12)\)  \(\approx\)  \(0.5562770662\)
\(L(1)\)   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

−14.35536971955271907032549400805, −12.60208685449650505161038028171, −11.85638185382237433096216740178, −10.25281064866352122952133148856, −9.285612671148327135423610248964, −8.142770115043182775628926715441, −7.75107999141018477134556915583, −6.40277395728024042605463354872, −4.42935228777675251968558759391, −3.33065637603436017905043258494, 2.33705050726053221832002406374, 3.20872019042703496496577222133, 5.49083214680502202454568705016, 7.19808119128069739656818853077, 8.159577564624528159149562240696, 9.180058689512610881312014020254, 10.17226644292946765125862866461, 11.50491176724106319037079544844, 11.93830801040101983817239437555, 13.11171005371797173645660263897

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