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Properties

Label 2-931-7.4-c1-0-9
Degree $2$
Conductor $931$
Sign $0.991 - 0.126i$
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  + (−0.809 − 1.40i)2-s + (−0.190 + 0.330i)3-s + (−0.309 + 0.535i)4-s + (−0.5 − 0.866i)5-s + 0.618·6-s − 2.23·8-s + (1.42 + 2.47i)9-s + (−0.809 + 1.40i)10-s + (−0.309 + 0.535i)11-s + (−0.118 − 0.204i)12-s − 13-s + 0.381·15-s + (2.42 + 4.20i)16-s + (−1.92 + 3.33i)17-s + (2.30 − 3.99i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.572 − 0.990i)2-s + (−0.110 + 0.190i)3-s + (−0.154 + 0.267i)4-s + (−0.223 − 0.387i)5-s + 0.252·6-s − 0.790·8-s + (0.475 + 0.823i)9-s + (−0.255 + 0.443i)10-s + (−0.0931 + 0.161i)11-s + (−0.0340 − 0.0590i)12-s − 0.277·13-s + 0.0986·15-s + (0.606 + 1.05i)16-s + (−0.467 + 0.809i)17-s + (0.544 − 0.942i)18-s + (0.114 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821129 + 0.0521090i\)
\(L(\frac12)\) \(\approx\) \(0.821129 + 0.0521090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.809 + 1.40i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.190 - 0.330i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.309 - 0.535i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (1.92 - 3.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.73 - 4.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + (4.78 - 8.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 2.18i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 - 8.47T + 43T^{2} \)
47 \( 1 + (3.73 + 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.42 - 4.20i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.88 - 6.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.97 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 2.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.70T + 71T^{2} \)
73 \( 1 + (-5.66 + 9.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.23 - 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.14T + 83T^{2} \)
89 \( 1 + (1.38 + 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.47T + 97T^{2} \)
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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

−10.33532768498860287180697070122, −9.373406133579702288306223788124, −8.683725316162921673170741830734, −7.79494211871461167081600457629, −6.76599494588248512135716149132, −5.58031864983934324239508622119, −4.67565898809586399007095569632, −3.56264718351669411897165355935, −2.32150406027769808619270819949, −1.28409361608178823589123460168, 0.51498057059916952385696235195, 2.58180467232992135455490088068, 3.68926534652449666169724480242, 4.99251417831809499031691415130, 6.12908998579813106363067015156, 6.81870239267038427089465983439, 7.37886996774784717771185948281, 8.191417234566691470177833413392, 9.273507589853751449107440797291, 9.554724965280288437475050966426

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