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Properties

Label 2-231-77.10-c1-0-8
Degree $2$
Conductor $231$
Sign $0.444 + 0.895i$
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  + (−0.931 + 0.537i)2-s + (−0.866 − 0.5i)3-s + (−0.421 + 0.729i)4-s + (−2.91 + 1.68i)5-s + 1.07·6-s + (0.879 − 2.49i)7-s − 3.05i·8-s + (0.499 + 0.866i)9-s + (1.80 − 3.13i)10-s + (1.21 − 3.08i)11-s + (0.729 − 0.421i)12-s + 4.53·13-s + (0.522 + 2.79i)14-s + 3.36·15-s + (0.802 + 1.38i)16-s + (−0.880 + 1.52i)17-s + ⋯
L(s)  = 1  + (−0.658 + 0.380i)2-s + (−0.499 − 0.288i)3-s + (−0.210 + 0.364i)4-s + (−1.30 + 0.752i)5-s + 0.439·6-s + (0.332 − 0.943i)7-s − 1.08i·8-s + (0.166 + 0.288i)9-s + (0.572 − 0.990i)10-s + (0.366 − 0.930i)11-s + (0.210 − 0.121i)12-s + 1.25·13-s + (0.139 + 0.747i)14-s + 0.868·15-s + (0.200 + 0.347i)16-s + (−0.213 + 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351190 - 0.217855i\)
\(L(\frac12)\) \(\approx\) \(0.351190 - 0.217855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.879 + 2.49i)T \)
11 \( 1 + (-1.21 + 3.08i)T \)
good2 \( 1 + (0.931 - 0.537i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.91 - 1.68i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.53T + 13T^{2} \)
17 \( 1 + (0.880 - 1.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.52 + 6.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.67 + 2.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 + (2.78 + 1.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.52 - 7.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.67T + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + (4.51 - 2.60i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.52 + 9.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.91 - 2.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.72 + 2.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.151 - 0.262i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.04T + 71T^{2} \)
73 \( 1 + (5.07 - 8.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.684 + 0.395i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.19T + 83T^{2} \)
89 \( 1 + (-7.44 + 4.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.01iT - 97T^{2} \)
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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

−11.64922890126358930607289484967, −11.16248909783669986345109714812, −10.30704770961071502745758481667, −8.612103426793367258909485779694, −8.129770961168062260084743143513, −7.03545195969780799839995760265, −6.44819646283984698844035605777, −4.30798526265481331332560196348, −3.52731029075905985736837779768, −0.48891420200029042518876076144, 1.54782374525725106801023508996, 3.94632093006159607476618676150, 4.97234527440548590386726204347, 6.05989549499932090579346466160, 7.80331499240838188008107359675, 8.647487687318334928727128099242, 9.332673597719403673696937014997, 10.53738473326071736785292419658, 11.44205507952627257084343034368, 12.00999528001176329735624519051

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