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Properties

Label 2-931-133.25-c1-0-48
Degree $2$
Conductor $931$
Sign $0.701 - 0.712i$
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  + (2.37 + 0.866i)2-s + (0.5 − 0.419i)3-s + (3.37 + 2.83i)4-s + (1.03 − 0.866i)5-s + (1.55 − 0.565i)6-s + (3.05 + 5.28i)8-s + (−0.446 + 2.53i)9-s + (3.20 − 1.16i)10-s − 1.18·11-s + 2.87·12-s + (2.55 − 0.929i)13-s + (0.152 − 0.866i)15-s + (1.15 + 6.53i)16-s + (−0.673 − 3.82i)17-s + (−3.25 + 5.64i)18-s + (−3.29 + 2.84i)19-s + ⋯
L(s)  = 1  + (1.68 + 0.612i)2-s + (0.288 − 0.242i)3-s + (1.68 + 1.41i)4-s + (0.461 − 0.387i)5-s + (0.634 − 0.230i)6-s + (1.07 + 1.86i)8-s + (−0.148 + 0.844i)9-s + (1.01 − 0.368i)10-s − 0.357·11-s + 0.831·12-s + (0.708 − 0.257i)13-s + (0.0394 − 0.223i)15-s + (0.288 + 1.63i)16-s + (−0.163 − 0.926i)17-s + (−0.768 + 1.33i)18-s + (−0.756 + 0.653i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.41366 + 1.84799i\)
\(L(\frac12)\) \(\approx\) \(4.41366 + 1.84799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.29 - 2.84i)T \)
good2 \( 1 + (-2.37 - 0.866i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.5 + 0.419i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (-1.03 + 0.866i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + 1.18T + 11T^{2} \)
13 \( 1 + (-2.55 + 0.929i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.673 + 3.82i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-4.75 + 1.73i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (3.56 + 2.99i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.91 + 3.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.05 - 3.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.38 + 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.51 + 8.57i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.0996 - 0.565i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-2.25 - 1.89i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.683 - 3.87i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.24 - 1.54i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-3.65 + 1.32i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (1.20 + 6.83i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (4.69 - 3.93i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-1.70 - 9.65i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-6.15 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.85 - 1.55i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (5.64 - 4.73i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.44143732239475680652407047356, −9.106303361511516214893301886831, −8.179806825220322129238634584503, −7.41321096256443568703394815124, −6.57168725034380778398903319574, −5.54311676238143322863475892584, −5.12916972054239303599136828203, −4.05263689369441130249358002214, −2.97560771578550958603848587635, −1.97222077841127596800619999421, 1.65791361733350458346731911503, 2.82608031925529680285823345183, 3.57322534658774273304568819460, 4.44094648145481443018253330815, 5.45842300466571669150932935404, 6.34621045671887711205419630336, 6.81297558887104129685506630331, 8.395879939102036509353381376794, 9.326504126667342027400617275413, 10.34632601545170703636724847312

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