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Properties

Label 2-931-133.12-c1-0-13
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  − 2.18i·2-s + (0.5 − 0.866i)3-s − 2.79·4-s + 2.64i·5-s + (−1.89 − 1.09i)6-s + 1.73i·8-s + (1 + 1.73i)9-s + 5.79·10-s + (−1.39 + 2.41i)12-s + (−2.29 + 3.96i)13-s + (2.29 + 1.32i)15-s − 1.79·16-s + (5.29 + 3.05i)17-s + (3.79 − 2.18i)18-s + (−0.5 + 4.33i)19-s − 7.38i·20-s + ⋯
L(s)  = 1  − 1.54i·2-s + (0.288 − 0.499i)3-s − 1.39·4-s + 1.18i·5-s + (−0.773 − 0.446i)6-s + 0.612i·8-s + (0.333 + 0.577i)9-s + 1.83·10-s + (−0.402 + 0.697i)12-s + (−0.635 + 1.10i)13-s + (0.591 + 0.341i)15-s − 0.447·16-s + (1.28 + 0.740i)17-s + (0.893 − 0.515i)18-s + (−0.114 + 0.993i)19-s − 1.65i·20-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: $\chi_{931} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45149 - 0.507800i\)
\(L(\frac12)\) \(\approx\) \(1.45149 - 0.507800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 4.33i)T \)
good2 \( 1 + 2.18iT - 2T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.64iT - 5T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.29 - 3.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.29 - 3.05i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.29 - 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.08 + 3.51i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.708 + 1.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.708 + 1.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.08 + 1.77i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.8 + 7.43i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 - 11.4iT - 67T^{2} \)
71 \( 1 + (3.70 - 2.14i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.87 + 2.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.00iT - 79T^{2} \)
83 \( 1 - 7.11iT - 83T^{2} \)
89 \( 1 + (0.708 + 1.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.87 + 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05656101949927601455864800179, −9.667046605573357396407222443222, −8.401900795003378236878305107313, −7.42708815082133511983914992912, −6.80016341086038388218310434249, −5.48428944044355174309107466462, −4.10241872605183016279005464292, −3.34428485174907266171943179274, −2.28935659061827311276455838905, −1.58055815963887710377193644593, 0.74095260136762910050635101102, 2.99534932384014725542852177741, 4.38028978645205122832058495774, 5.06923387558884919706755506059, 5.67611477024837533187791924600, 6.89281544046271964693019830589, 7.59250998877246035755372214334, 8.436615339878000496342245763254, 9.139778208336327212618522568341, 9.636841449045347063270167846954

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