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Properties

Label 2-931-133.93-c1-0-23
Degree $2$
Conductor $931$
Sign $0.781 + 0.623i$
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  + (−1.03 − 0.866i)2-s + (−0.5 + 2.83i)3-s + (−0.0320 − 0.181i)4-s + (0.152 − 0.866i)5-s + (2.97 − 2.49i)6-s + (−1.47 + 2.54i)8-s + (−4.97 − 1.80i)9-s + (−0.907 + 0.761i)10-s − 2.22·11-s + 0.532·12-s + (1.97 − 1.65i)13-s + (2.37 + 0.866i)15-s + (3.37 − 1.22i)16-s + (−0.439 + 0.160i)17-s + (3.56 + 6.17i)18-s + (1.52 − 4.08i)19-s + ⋯
L(s)  = 1  + (−0.729 − 0.612i)2-s + (−0.288 + 1.63i)3-s + (−0.0160 − 0.0909i)4-s + (0.0682 − 0.387i)5-s + (1.21 − 1.01i)6-s + (−0.520 + 0.901i)8-s + (−1.65 − 0.603i)9-s + (−0.287 + 0.240i)10-s − 0.671·11-s + 0.153·12-s + (0.546 − 0.458i)13-s + (0.614 + 0.223i)15-s + (0.844 − 0.307i)16-s + (−0.106 + 0.0388i)17-s + (0.840 + 1.45i)18-s + (0.348 − 0.937i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702342 - 0.245885i\)
\(L(\frac12)\) \(\approx\) \(0.702342 - 0.245885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 4.08i)T \)
good2 \( 1 + (1.03 + 0.866i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (0.5 - 2.83i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-0.152 + 0.866i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 + (-1.97 + 1.65i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.439 - 0.160i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.06 - 1.73i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.19 + 6.77i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.55 + 6.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.47 - 4.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.89 - 1.59i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.66 - 1.33i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-6.85 - 2.49i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.492 + 2.79i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-5.92 + 2.15i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-6.99 + 5.86i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.87 + 4.93i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-8.74 + 3.18i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.241 + 1.36i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-11.1 + 4.05i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.41 - 12.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.78 + 10.1i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.64 + 9.30i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.845822839890001649953503640334, −9.537951507334734932706451271243, −8.663526526717606112868714252738, −7.951126677946536282663303960153, −6.22775621622883224057500435618, −5.33126160822428658235672661328, −4.77199356730825145681315303433, −3.57896110957899088100490345680, −2.46212462301878721513365145561, −0.59177756046573222050693271225, 1.01045233274117805227602290997, 2.38295294554761517045444715893, 3.61256862871365227917880992398, 5.36894778139768199024277852383, 6.33498128819220615985347142658, 6.91760559610730655084960520787, 7.50147970024187753938890091460, 8.346041949031745898639393290413, 8.836748634149283804120173345665, 10.14092812988785259325089321576

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