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Properties

Label 2-231-11.9-c1-0-1
Degree $2$
Conductor $231$
Sign $0.253 - 0.967i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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  + (−0.0347 + 0.106i)2-s + (−0.809 + 0.587i)3-s + (1.60 + 1.16i)4-s + (0.327 + 1.00i)5-s + (−0.0347 − 0.106i)6-s + (0.809 + 0.587i)7-s + (−0.362 + 0.263i)8-s + (0.309 − 0.951i)9-s − 0.119·10-s + (−1.98 − 2.65i)11-s − 1.98·12-s + (−1.80 + 5.55i)13-s + (−0.0909 + 0.0661i)14-s + (−0.858 − 0.623i)15-s + (1.21 + 3.73i)16-s + (0.866 + 2.66i)17-s + ⋯
L(s)  = 1  + (−0.0245 + 0.0756i)2-s + (−0.467 + 0.339i)3-s + (0.803 + 0.584i)4-s + (0.146 + 0.451i)5-s + (−0.0141 − 0.0436i)6-s + (0.305 + 0.222i)7-s + (−0.128 + 0.0932i)8-s + (0.103 − 0.317i)9-s − 0.0377·10-s + (−0.597 − 0.801i)11-s − 0.573·12-s + (−0.500 + 1.54i)13-s + (−0.0243 + 0.0176i)14-s + (−0.221 − 0.161i)15-s + (0.303 + 0.933i)16-s + (0.210 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.253 - 0.967i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.987789 + 0.762427i\)
\(L(\frac12)\) \(\approx\) \(0.987789 + 0.762427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (1.98 + 2.65i)T \)
good2 \( 1 + (0.0347 - 0.106i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.327 - 1.00i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.80 - 5.55i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.866 - 2.66i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.88 + 2.09i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.72T + 23T^{2} \)
29 \( 1 + (6.95 + 5.05i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.538 + 1.65i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.83 + 2.78i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.96 - 3.60i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.25T + 43T^{2} \)
47 \( 1 + (-6.62 + 4.81i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.43 + 13.6i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.15 + 2.29i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.55 + 14.0i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 8.29T + 67T^{2} \)
71 \( 1 + (-3.59 - 11.0i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.39 - 2.46i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.10 + 6.47i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.360 - 1.10i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2.90T + 89T^{2} \)
97 \( 1 + (1.49 - 4.61i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.13182020799164244401410245851, −11.32264603119056158233719431091, −10.83190342747959610824823246013, −9.547438788016283934502237970915, −8.413190818424927074868004592357, −7.23681297898121849805441787779, −6.41045382618271855312901183260, −5.22035119039282280749299462413, −3.68955542039957253961553154446, −2.29709192346879545223094572797, 1.21318262706891756834934299029, 2.84286617326086480018842749357, 5.10663490406368196458413601257, 5.53942967214852638311793446442, 7.15657374619256554930073410363, 7.61599504832691758595523392036, 9.251540030733974633615175671530, 10.38314489231897087577319122095, 10.86350884054011338823955558165, 12.13583741973218977773242674948

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