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Properties

Label 2-231-11.4-c1-0-4
Degree $2$
Conductor $231$
Sign $0.198 - 0.980i$
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  + (2.14 + 1.55i)2-s + (0.309 − 0.951i)3-s + (1.55 + 4.78i)4-s + (−1.38 + 1.00i)5-s + (2.14 − 1.55i)6-s + (−0.309 − 0.951i)7-s + (−2.48 + 7.64i)8-s + (−0.809 − 0.587i)9-s − 4.54·10-s + (2.49 − 2.18i)11-s + 5.03·12-s + (−1.22 − 0.888i)13-s + (0.819 − 2.52i)14-s + (0.529 + 1.62i)15-s + (−9.11 + 6.62i)16-s + (1.93 − 1.40i)17-s + ⋯
L(s)  = 1  + (1.51 + 1.10i)2-s + (0.178 − 0.549i)3-s + (0.777 + 2.39i)4-s + (−0.619 + 0.450i)5-s + (0.875 − 0.636i)6-s + (−0.116 − 0.359i)7-s + (−0.878 + 2.70i)8-s + (−0.269 − 0.195i)9-s − 1.43·10-s + (0.752 − 0.658i)11-s + 1.45·12-s + (−0.339 − 0.246i)13-s + (0.219 − 0.674i)14-s + (0.136 + 0.420i)15-s + (−2.27 + 1.65i)16-s + (0.469 − 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.198 - 0.980i$
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: \((2,\ 231,\ (\ :1/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94654 + 1.59178i\)
\(L(\frac12)\) \(\approx\) \(1.94654 + 1.59178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.49 + 2.18i)T \)
good2 \( 1 + (-2.14 - 1.55i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.38 - 1.00i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (1.22 + 0.888i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.93 + 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.95 + 6.02i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.05T + 23T^{2} \)
29 \( 1 + (-2.57 - 7.93i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.88 + 3.54i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.30 + 7.10i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.55 - 4.78i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.02T + 43T^{2} \)
47 \( 1 + (1.70 - 5.25i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.04 + 3.66i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.45 - 10.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.1 - 8.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + (-11.5 + 8.42i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.441 + 1.36i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.43 + 3.94i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.74 - 3.45i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (9.74 + 7.08i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.65727290090779763611751897272, −11.82198940024247396705188323542, −11.05748653068286380991531301318, −9.079899600757828469555776908658, −7.80151045782068801323053170722, −7.21980308736630114503888079461, −6.37303146741017581969942520797, −5.23135816603338273259774819082, −3.88068083516941458396028694177, −2.99936027525386509026171348800, 1.92449870108089327212593371953, 3.54886807408444071510806044479, 4.23499493401124181567715221355, 5.28436135415380082605479434694, 6.39178933921996426084360814752, 8.114324347558166895822187412431, 9.645081120849423737509484901593, 10.16787909784886881086031996824, 11.46277999507747239547154631101, 12.17232686438440665016497077842

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