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Properties

Label 2-931-133.123-c1-0-44
Degree $2$
Conductor $931$
Sign $0.729 + 0.684i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.03 + 0.866i)2-s + (0.5 + 2.83i)3-s + (−0.0320 + 0.181i)4-s + (−0.152 − 0.866i)5-s + (−2.97 − 2.49i)6-s + (−1.47 − 2.54i)8-s + (−4.97 + 1.80i)9-s + (0.907 + 0.761i)10-s − 2.22·11-s − 0.532·12-s + (−1.97 − 1.65i)13-s + (2.37 − 0.866i)15-s + (3.37 + 1.22i)16-s + (0.439 + 0.160i)17-s + (3.56 − 6.17i)18-s + (−1.52 − 4.08i)19-s + ⋯
L(s)  = 1  + (−0.729 + 0.612i)2-s + (0.288 + 1.63i)3-s + (−0.0160 + 0.0909i)4-s + (−0.0682 − 0.387i)5-s + (−1.21 − 1.01i)6-s + (−0.520 − 0.901i)8-s + (−1.65 + 0.603i)9-s + (0.287 + 0.240i)10-s − 0.671·11-s − 0.153·12-s + (−0.546 − 0.458i)13-s + (0.614 − 0.223i)15-s + (0.844 + 0.307i)16-s + (0.106 + 0.0388i)17-s + (0.840 − 1.45i)18-s + (−0.348 − 0.937i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.729 + 0.684i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.729 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0814426 - 0.0322430i\)
\(L(\frac12)\) \(\approx\) \(0.0814426 - 0.0322430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 + 4.08i)T \)
good2 \( 1 + (1.03 - 0.866i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-0.5 - 2.83i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (0.152 + 0.866i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 + (1.97 + 1.65i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.439 - 0.160i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.06 + 1.73i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.19 - 6.77i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.55 + 6.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.47 + 4.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.89 - 1.59i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.66 + 1.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (6.85 - 2.49i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (0.492 - 2.79i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (5.92 + 2.15i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (6.99 + 5.86i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.87 - 4.93i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-8.74 - 3.18i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.241 + 1.36i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-11.1 - 4.05i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (7.41 - 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.78 + 10.1i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.64 + 9.30i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.714365503749357377143892796745, −9.145864089802793022386001214021, −8.460401926557235468506277280469, −7.77215696483886760066856873725, −6.69608415672713069617748603848, −5.39625657995629602710296843668, −4.69991974713807700551515314768, −3.69115467550401404468117782246, −2.75186593102138810008308190176, −0.04934543021706651723839296075, 1.48882961232655279408848568665, 2.23790720582447701941842409105, 3.23984065081977509938873455065, 5.10874413211911435463253380599, 6.13140676075085387033777721632, 6.91945588528898428415892222121, 7.84811126560987543820429158739, 8.336365344029390594758267116985, 9.320593267874970528936740110097, 10.18983026823879812772924768792

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