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Properties

Label 2-231-11.9-c1-0-7
Degree $2$
Conductor $231$
Sign $0.882 + 0.469i$
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  + (0.118 − 0.363i)2-s + (0.809 − 0.587i)3-s + (1.5 + 1.08i)4-s + (−0.190 − 0.587i)5-s + (−0.118 − 0.363i)6-s + (−0.809 − 0.587i)7-s + (1.19 − 0.865i)8-s + (0.309 − 0.951i)9-s − 0.236·10-s + (3.04 − 1.31i)11-s + 1.85·12-s + (−1.07 + 3.30i)13-s + (−0.309 + 0.224i)14-s + (−0.5 − 0.363i)15-s + (0.972 + 2.99i)16-s + (−0.927 − 2.85i)17-s + ⋯
L(s)  = 1  + (0.0834 − 0.256i)2-s + (0.467 − 0.339i)3-s + (0.750 + 0.544i)4-s + (−0.0854 − 0.262i)5-s + (−0.0481 − 0.148i)6-s + (−0.305 − 0.222i)7-s + (0.421 − 0.305i)8-s + (0.103 − 0.317i)9-s − 0.0746·10-s + (0.918 − 0.396i)11-s + 0.535·12-s + (−0.297 + 0.915i)13-s + (−0.0825 + 0.0600i)14-s + (−0.129 − 0.0937i)15-s + (0.243 + 0.747i)16-s + (−0.224 − 0.691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.882 + 0.469i$
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: \((2,\ 231,\ (\ :1/2),\ 0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62324 - 0.405202i\)
\(L(\frac12)\) \(\approx\) \(1.62324 - 0.405202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 + 1.31i)T \)
good2 \( 1 + (-0.118 + 0.363i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.190 + 0.587i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.07 - 3.30i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.927 + 2.85i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.23 - 2.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.76T + 23T^{2} \)
29 \( 1 + (2.42 + 1.76i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.30 - 4.02i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.381 + 0.277i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.54 - 4.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-0.881 + 0.640i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.19 - 6.74i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.39 + 6.10i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.07 - 6.37i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.70T + 67T^{2} \)
71 \( 1 + (-2.39 - 7.38i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.30 + 1.67i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.0450 + 0.138i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.61 + 14.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + (-3.21 + 9.90i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.07128586724565053201718882654, −11.46352518425255255544963559723, −10.24654028086749711101845115251, −9.086110875265648189970230710875, −8.185461155209004286643589092630, −7.01272608907639050146532316387, −6.36968832710382459965484475667, −4.36772479626581962474302252876, −3.26044887810231908010556120804, −1.80881972884943938460824519444, 2.06207235870521909400671977218, 3.50764289190871087620009182506, 5.02911834213882559099577330579, 6.27475659065205344952102446760, 7.12224687267854619582669226270, 8.301452084864150850446351078094, 9.462790685113802914775328223156, 10.36048170162547716851051628720, 11.14522830049566416233003130613, 12.25262308095280143609540869035

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