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Properties

Label 2-231-11.4-c1-0-2
Degree $2$
Conductor $231$
Sign $-0.394 - 0.918i$
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  + (1.80 + 1.31i)2-s + (−0.309 + 0.951i)3-s + (0.927 + 2.85i)4-s + (−1.30 + 0.951i)5-s + (−1.80 + 1.31i)6-s + (0.309 + 0.951i)7-s + (−0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s − 3.61·10-s + (0.809 − 3.21i)11-s − 2.99·12-s + (1 + 0.726i)13-s + (−0.690 + 2.12i)14-s + (−0.499 − 1.53i)15-s + (0.809 − 0.587i)16-s + (1.5 − 1.08i)17-s + ⋯
L(s)  = 1  + (1.27 + 0.929i)2-s + (−0.178 + 0.549i)3-s + (0.463 + 1.42i)4-s + (−0.585 + 0.425i)5-s + (−0.738 + 0.536i)6-s + (0.116 + 0.359i)7-s + (−0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s − 1.14·10-s + (0.243 − 0.969i)11-s − 0.866·12-s + (0.277 + 0.201i)13-s + (−0.184 + 0.568i)14-s + (−0.129 − 0.397i)15-s + (0.202 − 0.146i)16-s + (0.363 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.394 - 0.918i$
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: \((2,\ 231,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13234 + 1.71871i\)
\(L(\frac12)\) \(\approx\) \(1.13234 + 1.71871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 3.21i)T \)
good2 \( 1 + (-1.80 - 1.31i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1 - 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.5 + 1.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.19 - 3.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + (1.85 + 5.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.54 + 4.75i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.736 - 2.26i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.97 + 9.14i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.70T + 43T^{2} \)
47 \( 1 + (1.38 - 4.25i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.09 + 6.60i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.381 + 1.17i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (10.4 - 7.60i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.0901 - 0.277i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.09 - 5.87i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.09 - 4.42i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + (-4.85 - 3.52i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.67915505073502603500358756017, −11.61103889826456752092112142722, −10.99008503284219463519804924266, −9.508633215205355930168058937092, −8.258712057840916865504512551967, −7.23884287980394428942887592653, −6.10142010350435782309384236173, −5.35074586395700497433580769644, −4.05302203030036290309211996989, −3.25359258629615319118096112760, 1.50658874267227019928295598953, 3.14212594990844433394571175096, 4.40650497033866931382246944218, 5.20582693833899828085690554349, 6.64319819542584380170959766054, 7.73411596900232356257757697342, 9.054329496331560862426876153933, 10.53876858173848507002407439355, 11.22352336400187125716428542857, 12.14648575135531499098428175945

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