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Properties

Label 2-931-133.132-c1-0-47
Degree $2$
Conductor $931$
Sign $-0.529 + 0.848i$
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  − 2.27i·2-s + 2.74·3-s − 3.17·4-s + 1.16i·5-s − 6.23i·6-s + 2.67i·8-s + 4.51·9-s + 2.66·10-s + 4.83·11-s − 8.71·12-s − 3.16·13-s + 3.20i·15-s − 0.258·16-s − 3.82i·17-s − 10.2i·18-s + (−3.29 − 2.85i)19-s + ⋯
L(s)  = 1  − 1.60i·2-s + 1.58·3-s − 1.58·4-s + 0.522i·5-s − 2.54i·6-s + 0.947i·8-s + 1.50·9-s + 0.841·10-s + 1.45·11-s − 2.51·12-s − 0.876·13-s + 0.827i·15-s − 0.0646·16-s − 0.928i·17-s − 2.42i·18-s + (−0.755 − 0.655i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.529 + 0.848i$
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: \((2,\ 931,\ (\ :1/2),\ -0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27598 - 2.29948i\)
\(L(\frac12)\) \(\approx\) \(1.27598 - 2.29948i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.29 + 2.85i)T \)
good2 \( 1 + 2.27iT - 2T^{2} \)
3 \( 1 - 2.74T + 3T^{2} \)
5 \( 1 - 1.16iT - 5T^{2} \)
11 \( 1 - 4.83T + 11T^{2} \)
13 \( 1 + 3.16T + 13T^{2} \)
17 \( 1 + 3.82iT - 17T^{2} \)
23 \( 1 - 7.97T + 23T^{2} \)
29 \( 1 + 7.81iT - 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 - 2.56iT - 37T^{2} \)
41 \( 1 - 4.59T + 41T^{2} \)
43 \( 1 + 6.25T + 43T^{2} \)
47 \( 1 + 1.49iT - 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 14.3iT - 61T^{2} \)
67 \( 1 - 4.81iT - 67T^{2} \)
71 \( 1 - 9.62iT - 71T^{2} \)
73 \( 1 - 3.07iT - 73T^{2} \)
79 \( 1 + 4.04iT - 79T^{2} \)
83 \( 1 + 4.15iT - 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 0.0766T + 97T^{2} \)
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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

−9.677500702510316831167855599560, −9.185234953410701033061042303354, −8.636480550821102100161031630407, −7.36230350415964589194617681747, −6.66823486028144421406143529144, −4.68028357857577144360783396506, −3.99420130668790616070072597634, −2.79411251413140487912298540671, −2.65856844687927600010347236852, −1.22474183734820694382941048730, 1.67535060978812639640517213970, 3.24945944652326113656645618695, 4.28607896602229020603012688930, 5.08480857221743956771396211324, 6.43844951704688403777284013921, 7.00304174058114007027077909262, 7.929851237489111886379824320762, 8.594400850134925423090129095336, 9.052505566786046113688146637015, 9.672014603519135723394831838445

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