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Properties

Label 2-231-11.5-c1-0-0
Degree $2$
Conductor $231$
Sign $0.999 + 0.0109i$
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  + (−0.739 − 2.27i)2-s + (−0.809 − 0.587i)3-s + (−3.01 + 2.19i)4-s + (−1.21 + 3.74i)5-s + (−0.739 + 2.27i)6-s + (0.809 − 0.587i)7-s + (3.35 + 2.43i)8-s + (0.309 + 0.951i)9-s + 9.42·10-s + (2.04 + 2.61i)11-s + 3.73·12-s + (−0.926 − 2.85i)13-s + (−1.93 − 1.40i)14-s + (3.18 − 2.31i)15-s + (0.759 − 2.33i)16-s + (−2.04 + 6.28i)17-s + ⋯
L(s)  = 1  + (−0.523 − 1.60i)2-s + (−0.467 − 0.339i)3-s + (−1.50 + 1.09i)4-s + (−0.544 + 1.67i)5-s + (−0.302 + 0.929i)6-s + (0.305 − 0.222i)7-s + (1.18 + 0.861i)8-s + (0.103 + 0.317i)9-s + 2.98·10-s + (0.615 + 0.788i)11-s + 1.07·12-s + (−0.257 − 0.791i)13-s + (−0.517 − 0.376i)14-s + (0.822 − 0.597i)15-s + (0.189 − 0.584i)16-s + (−0.494 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.999 + 0.0109i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.999 + 0.0109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558566 - 0.00306136i\)
\(L(\frac12)\) \(\approx\) \(0.558566 - 0.00306136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 2.61i)T \)
good2 \( 1 + (0.739 + 2.27i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (1.21 - 3.74i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.926 + 2.85i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.04 - 6.28i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.77 - 3.46i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.02T + 23T^{2} \)
29 \( 1 + (-1.23 + 0.893i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.61 - 8.05i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.491 + 0.357i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.37 - 1.00i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.23T + 43T^{2} \)
47 \( 1 + (-3.88 - 2.82i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.89 + 5.82i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.04 - 3.66i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.645 - 1.98i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.599T + 67T^{2} \)
71 \( 1 + (-0.435 + 1.34i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.73 - 4.16i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.316 + 0.973i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.954 + 2.93i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.48T + 89T^{2} \)
97 \( 1 + (-0.788 - 2.42i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.96363748683284645335933695735, −11.19067610177940392997157381711, −10.34444964120030019934648226991, −10.02031580340827646979905985893, −8.298025333064966913401826123657, −7.39886076326474664729982781868, −6.21560272949170898451810323266, −4.18069788423393273675194789026, −3.17344754201496678159376743496, −1.77770534822738487341868455946, 0.60371171088443339071375612764, 4.37095776312934886049849863903, 5.06054677623181478793511104964, 6.02862770375254702302302877368, 7.29077201193851289877820383861, 8.245842137067185269914416505502, 9.175495761976068395134387075985, 9.486987693412292874842933620497, 11.56381984049782260591373347440, 11.92327504523196146431494115450

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