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Properties

Label 2-931-133.87-c0-0-1
Degree $2$
Conductor $931$
Sign $0.623 + 0.781i$
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  + 2-s + (0.866 − 0.5i)3-s − i·5-s + (0.866 − 0.5i)6-s − 8-s − i·10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s − 16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (−0.5 + 0.866i)23-s + (−0.866 + 0.5i)24-s + (0.866 − 0.5i)26-s + i·27-s + ⋯
L(s)  = 1  + 2-s + (0.866 − 0.5i)3-s − i·5-s + (0.866 − 0.5i)6-s − 8-s − i·10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)15-s − 16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (−0.5 + 0.866i)23-s + (−0.866 + 0.5i)24-s + (0.866 − 0.5i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.623 + 0.781i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.623 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.836735353\)
\(L(\frac12)\) \(\approx\) \(1.836735353\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
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 \( 1 - T + T^{2} \)
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 \( 1 - T + T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00709670894929237267132985761, −9.048028168906958751538310402713, −8.442661125866059687325842608407, −7.86698758689669587943307718338, −6.57264341231224077857680132557, −5.53011242451218015388325022729, −4.92363942433636369361111497302, −3.75305611721712831070435243737, −3.00479793077426492278871213494, −1.52381111519917848447664007832, 2.40581941869445838551430407125, 3.34293532111031811754190962272, 3.90302025822603871370413067572, 4.86222757565734195238070820249, 6.25794407683061799466283437696, 6.52218581197446863637031872744, 8.093116502308640993661135637002, 8.665581847532356952401150592026, 9.588769826292182471516726914103, 10.34033431855747920872513222646

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