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Properties

Label 2-931-133.68-c0-0-0
Degree $2$
Conductor $931$
Sign $0.962 - 0.272i$
Analytic cond. $0.464629$
Root an. cond. $0.681637$
Motivic weight $0$
Arithmetic yes
Rational no
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 + i·3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s − 8-s + (0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + i·17-s + (0.866 + 0.5i)19-s + 23-s − i·24-s + (−0.866 − 0.499i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + i·3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s − 8-s + (0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + i·17-s + (0.866 + 0.5i)19-s + 23-s − i·24-s + (−0.866 − 0.499i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7135766312\)
\(L(\frac12)\) \(\approx\) \(0.7135766312\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 - iT - T^{2} \)
5 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - iT - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + iT - T^{2} \)
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 \( 1 + iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43676262681067968055608137782, −9.731789625085636846671381449877, −8.883899026118061759349448557131, −8.087318345926855616247234391069, −6.97387248822256457515986040301, −5.91838539305806821428711560719, −4.83793345792158278743608965280, −3.49483453170177207935044155537, −3.30682720694854151566331497045, −1.46401039924033847463452780961, 0.948575539220244715818027798531, 2.71183074506747621594498340452, 3.97796069901549526153324061350, 5.19908999135408060994142871358, 6.36620652026159446138196942441, 7.08697363604737370016753112194, 7.54383511545847247837709464738, 8.426873909798169770306747984297, 8.922767286655287649361857608899, 9.965323104378623187342984000636

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