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Properties

Label 2-931-133.132-c1-0-18
Degree $2$
Conductor $931$
Sign $-0.837 + 0.546i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.837 + 0.546i$
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: \((2,\ 931,\ (\ :1/2),\ -0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625106 - 2.10140i\)
\(L(\frac12)\) \(\approx\) \(0.625106 - 2.10140i\)
\(L(\frac{3}{2})\) 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

−10.22823694361571676564836426428, −9.266581397936778730823367455666, −8.606013449231150149799402588018, −7.85430065796877150541102852178, −7.31991663453541327007026218779, −6.53554276911849826333679480771, −5.80164075004137125432278483492, −4.21965095560775952869174620035, −3.54342098995431586424527035003, −2.39219299014501474576836185015, 0.904690098042732659984551802336, 1.96692003581060395920149933564, 2.85017224297408840475901365872, 3.96902445448594295424595364197, 4.45634689714872937384460443667, 5.58687018091029033787715757323, 7.80538582857125230311148499529, 8.284420251998593074155653193151, 9.105167850128644515860698020217, 9.409036949630981127576422490281

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