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Properties

Label 2-931-133.122-c1-0-27
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  − 0.456i·2-s + (0.5 + 0.866i)3-s + 1.79·4-s + 2.64i·5-s + (0.395 − 0.228i)6-s − 1.73i·8-s + (1 − 1.73i)9-s + 1.20·10-s + (0.895 + 1.55i)12-s + (2.29 + 3.96i)13-s + (−2.29 + 1.32i)15-s + 2.79·16-s + (0.708 − 0.409i)17-s + (−0.791 − 0.456i)18-s + (−0.5 − 4.33i)19-s + 4.73i·20-s + ⋯
L(s)  = 1  − 0.323i·2-s + (0.288 + 0.499i)3-s + 0.895·4-s + 1.18i·5-s + (0.161 − 0.0932i)6-s − 0.612i·8-s + (0.333 − 0.577i)9-s + 0.382·10-s + (0.258 + 0.447i)12-s + (0.635 + 1.10i)13-s + (−0.591 + 0.341i)15-s + 0.697·16-s + (0.171 − 0.0992i)17-s + (−0.186 − 0.107i)18-s + (−0.114 − 0.993i)19-s + 1.05i·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: $\chi_{931} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17581 + 0.761200i\)
\(L(\frac12)\) \(\approx\) \(2.17581 + 0.761200i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.456iT - 2T^{2} \)
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 \( 1 + (-0.708 + 0.409i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.29 - 3.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.08 + 1.77i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.29 - 9.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.29 - 9.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.08 + 3.51i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.0953iT - 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.873 + 0.504i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 - 4.47iT - 67T^{2} \)
71 \( 1 + (8.29 + 4.78i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.87 + 5.70i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + (5.29 - 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.87 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.03710292581934112750175034795, −9.717827903052091423608739204619, −8.577364278706788198003258800927, −7.43123004972601606001232705204, −6.61566336864795532869899665196, −6.27795248189328684026666576749, −4.60381574344064119662862737040, −3.49606170177059564599093821510, −2.91339367268561641232129138104, −1.61936922770055403769122429138, 1.19309507118587372223445252685, 2.19815339053937754954280242905, 3.53362184455892862276963282896, 4.91389224542900367944369549959, 5.67193131752277319620668820673, 6.60534783009873762238511844959, 7.59070279585406147885311831175, 8.263304076412131796755603025452, 8.686953169389278327385328010333, 10.23407604541831034161994698890

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