Properties

Label 2-231-11.4-c1-0-8
Degree $2$
Conductor $231$
Sign $0.530 + 0.847i$
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  + (0.5 + 0.363i)2-s + (0.309 − 0.951i)3-s + (−0.5 − 1.53i)4-s + (0.309 − 0.224i)5-s + (0.5 − 0.363i)6-s + (0.309 + 0.951i)7-s + (0.690 − 2.12i)8-s + (−0.809 − 0.587i)9-s + 0.236·10-s + (0.309 − 3.30i)11-s − 1.61·12-s + (0.809 + 0.587i)13-s + (−0.190 + 0.587i)14-s + (−0.118 − 0.363i)15-s + (−1.49 + 1.08i)16-s + (3.42 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.256i)2-s + (0.178 − 0.549i)3-s + (−0.250 − 0.769i)4-s + (0.138 − 0.100i)5-s + (0.204 − 0.148i)6-s + (0.116 + 0.359i)7-s + (0.244 − 0.751i)8-s + (−0.269 − 0.195i)9-s + 0.0746·10-s + (0.0931 − 0.995i)11-s − 0.467·12-s + (0.224 + 0.163i)13-s + (−0.0510 + 0.157i)14-s + (−0.0304 − 0.0937i)15-s + (−0.374 + 0.272i)16-s + (0.831 − 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30393 - 0.722542i\)
\(L(\frac12)\) \(\approx\) \(1.30393 - 0.722542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.309 + 3.30i)T \)
good2 \( 1 + (-0.5 - 0.363i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.309 + 0.224i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.42 + 2.48i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.23T + 23T^{2} \)
29 \( 1 + (-2.07 - 6.37i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.28 - 6.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.14 - 6.60i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.57 + 4.84i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (2.26 - 6.96i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.16 + 4.47i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.28 + 3.94i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.66 + 3.38i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.23T + 67T^{2} \)
71 \( 1 + (-6.04 + 4.39i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.57 - 10.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.78 + 6.37i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.85 + 3.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.38T + 89T^{2} \)
97 \( 1 + (-13.7 - 9.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.15467920678847916414694603999, −11.17819590376084181314161693640, −10.04723551455530696745302389127, −9.067924828755025971252950313976, −8.102721981548189518230283964633, −6.75115014833070283486044015618, −5.87858042809533219002491134317, −4.92481144066176343642685981304, −3.24214224253041763523208463276, −1.29418741503686158838866551520, 2.44187535877568199674217879546, 3.88477580402045776520100038064, 4.59991917181909407194680261848, 6.08968059892339447994911850108, 7.64941458243845030308416010526, 8.288302585936006703225372111740, 9.645623347340596357785070450316, 10.33243737156570864560455349708, 11.58329586694541224946436845244, 12.29580061987095941937894623793

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