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Properties

Label 2-931-133.132-c1-0-20
Degree $2$
Conductor $931$
Sign $-0.965 - 0.260i$
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  − 2.63i·2-s − 2.79·3-s − 4.93·4-s + 3.67i·5-s + 7.36i·6-s + 7.72i·8-s + 4.83·9-s + 9.68·10-s − 1.26·11-s + 13.8·12-s − 2.26·13-s − 10.2i·15-s + 10.4·16-s + 5.47i·17-s − 12.7i·18-s + (1.78 − 3.97i)19-s + ⋯
L(s)  = 1  − 1.86i·2-s − 1.61·3-s − 2.46·4-s + 1.64i·5-s + 3.00i·6-s + 2.73i·8-s + 1.61·9-s + 3.06·10-s − 0.380·11-s + 3.98·12-s − 0.628·13-s − 2.65i·15-s + 2.61·16-s + 1.32i·17-s − 2.99i·18-s + (0.408 − 0.912i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.965 - 0.260i$
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: \((2,\ 931,\ (\ :1/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0311353 + 0.234627i\)
\(L(\frac12)\) \(\approx\) \(0.0311353 + 0.234627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 3.97i)T \)
good2 \( 1 + 2.63iT - 2T^{2} \)
3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 - 3.67iT - 5T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 - 5.47iT - 17T^{2} \)
23 \( 1 + 2.85T + 23T^{2} \)
29 \( 1 + 3.01iT - 29T^{2} \)
31 \( 1 + 0.0700T + 31T^{2} \)
37 \( 1 - 2.62iT - 37T^{2} \)
41 \( 1 + 8.52T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 5.31iT - 47T^{2} \)
53 \( 1 + 2.18iT - 53T^{2} \)
59 \( 1 + 4.02T + 59T^{2} \)
61 \( 1 + 3.89iT - 61T^{2} \)
67 \( 1 + 1.35iT - 67T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 - 7.37iT - 79T^{2} \)
83 \( 1 - 2.07iT - 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−10.27559389572797687095035204133, −9.439140781329438863704137958000, −7.976211428741838829823786680104, −6.85594210645422722381790832799, −5.97113540569280718337378397227, −4.99937977760286355372649512686, −4.00977906980067708051265453914, −2.97075123096831606514679301947, −1.90931143858467001524133944599, −0.18706745206342698057954134339, 0.937484843562142559168974778615, 4.23330654031710619228824775613, 4.90963500719316553024021079048, 5.43868785352618842641550448197, 5.96052769753326357447140904939, 7.08942947833177297560604848223, 7.71221274733750368235899069485, 8.666577955637648892617610735339, 9.485459063362296166739855191429, 10.15737278360552771868731298181

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