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Properties

Label 2-931-133.132-c1-0-49
Degree $2$
Conductor $931$
Sign $0.968 - 0.248i$
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  + 1.38i·2-s + 2.17·3-s + 0.0839·4-s − 4.26i·5-s + 3.01i·6-s + 2.88i·8-s + 1.74·9-s + 5.90·10-s + 0.143·11-s + 0.182·12-s + 2.01·13-s − 9.30i·15-s − 3.82·16-s − 2.60i·17-s + 2.42i·18-s + (3.40 − 2.71i)19-s + ⋯
L(s)  = 1  + 0.978i·2-s + 1.25·3-s + 0.0419·4-s − 1.90i·5-s + 1.23i·6-s + 1.01i·8-s + 0.582·9-s + 1.86·10-s + 0.0431·11-s + 0.0528·12-s + 0.559·13-s − 2.40i·15-s − 0.956·16-s − 0.631i·17-s + 0.570i·18-s + (0.782 − 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70007 + 0.340885i\)
\(L(\frac12)\) \(\approx\) \(2.70007 + 0.340885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.40 + 2.71i)T \)
good2 \( 1 - 1.38iT - 2T^{2} \)
3 \( 1 - 2.17T + 3T^{2} \)
5 \( 1 + 4.26iT - 5T^{2} \)
11 \( 1 - 0.143T + 11T^{2} \)
13 \( 1 - 2.01T + 13T^{2} \)
17 \( 1 + 2.60iT - 17T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 - 7.53iT - 29T^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
37 \( 1 + 9.99iT - 37T^{2} \)
41 \( 1 - 1.27T + 41T^{2} \)
43 \( 1 - 1.66T + 43T^{2} \)
47 \( 1 - 7.99iT - 47T^{2} \)
53 \( 1 + 5.72iT - 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 6.32iT - 61T^{2} \)
67 \( 1 - 5.56iT - 67T^{2} \)
71 \( 1 - 4.99iT - 71T^{2} \)
73 \( 1 - 3.74iT - 73T^{2} \)
79 \( 1 + 1.47iT - 79T^{2} \)
83 \( 1 - 2.80iT - 83T^{2} \)
89 \( 1 + 2.80T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−9.337532899012260934483057272514, −9.057009621043092773358332915727, −8.497170313254850144106070367521, −7.67138331454179421020147579790, −7.00885728150525175842702884967, −5.52135508479546202519759104860, −5.12345882215598785128530916208, −3.89595366109124324488947246658, −2.60539158587622588710072919597, −1.25994947065003288354548555144, 1.76909882486504260754728854072, 2.71696550695514850781900828398, 3.30429901686141893113445548064, 3.90424700257900174094481010932, 5.92532277361698817042637553725, 6.82896342846658900035728106382, 7.49853988579241557593240335162, 8.392031049497141674544836992223, 9.575932473431372708248158139335, 10.02804173357069581495788178012

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