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Properties

Label 2-231-11.4-c1-0-9
Degree $2$
Conductor $231$
Sign $0.940 + 0.339i$
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  + (1.07 + 0.783i)2-s + (0.309 − 0.951i)3-s + (−0.0686 − 0.211i)4-s + (0.706 − 0.513i)5-s + (1.07 − 0.783i)6-s + (−0.309 − 0.951i)7-s + (0.915 − 2.81i)8-s + (−0.809 − 0.587i)9-s + 1.16·10-s + (1.79 + 2.78i)11-s − 0.222·12-s + (−0.0319 − 0.0232i)13-s + (0.412 − 1.26i)14-s + (−0.269 − 0.830i)15-s + (2.83 − 2.06i)16-s + (−1.65 + 1.20i)17-s + ⋯
L(s)  = 1  + (0.762 + 0.554i)2-s + (0.178 − 0.549i)3-s + (−0.0343 − 0.105i)4-s + (0.315 − 0.229i)5-s + (0.440 − 0.319i)6-s + (−0.116 − 0.359i)7-s + (0.323 − 0.996i)8-s + (−0.269 − 0.195i)9-s + 0.368·10-s + (0.541 + 0.840i)11-s − 0.0641·12-s + (−0.00887 − 0.00644i)13-s + (0.110 − 0.338i)14-s + (−0.0696 − 0.214i)15-s + (0.709 − 0.515i)16-s + (−0.401 + 0.291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.940 + 0.339i$
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: \((2,\ 231,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85394 - 0.324710i\)
\(L(\frac12)\) \(\approx\) \(1.85394 - 0.324710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 - 2.78i)T \)
good2 \( 1 + (-1.07 - 0.783i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.706 + 0.513i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (0.0319 + 0.0232i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.65 - 1.20i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.75 - 5.41i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 + (1.44 + 4.44i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.05 + 4.39i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.57 - 11.0i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.90 - 5.84i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.79T + 43T^{2} \)
47 \( 1 + (-1.86 + 5.75i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.100 - 0.0731i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.43 + 13.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.11 - 4.44i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.85T + 67T^{2} \)
71 \( 1 + (-9.10 + 6.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.189 - 0.582i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.125 - 0.0912i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.21 - 5.24i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 5.17T + 89T^{2} \)
97 \( 1 + (6.42 + 4.66i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.62226341354355001580751646297, −11.37134276683981659942841919598, −10.05198745291161487320541970882, −9.307547675666673909479089377164, −7.88430999711631522065634844956, −6.85826215367546099581874684160, −6.08058571301127624480372415604, −4.88409802316470882149364909442, −3.72436087744063251370772324857, −1.60501865781524480009267735357, 2.47636004739886167054100950170, 3.50483226444589772463016159558, 4.69136359585841448597684424017, 5.74672395241108615117933882090, 7.14350131908222369989729572254, 8.711859401690109372367089883856, 9.167069018487953702676128974376, 10.81157256208703892642119783038, 11.15347866190196641609188823984, 12.33895633354544636927645062607

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