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Properties

Label 2-231-11.4-c1-0-7
Degree $2$
Conductor $231$
Sign $0.0694 - 0.997i$
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  + (1.93 + 1.40i)2-s + (−0.309 + 0.951i)3-s + (1.14 + 3.53i)4-s + (2.09 − 1.52i)5-s + (−1.93 + 1.40i)6-s + (−0.309 − 0.951i)7-s + (−1.26 + 3.89i)8-s + (−0.809 − 0.587i)9-s + 6.19·10-s + (−2.19 + 2.48i)11-s − 3.71·12-s + (−5.48 − 3.98i)13-s + (0.738 − 2.27i)14-s + (0.800 + 2.46i)15-s + (−1.91 + 1.38i)16-s + (2.54 − 1.84i)17-s + ⋯
L(s)  = 1  + (1.36 + 0.993i)2-s + (−0.178 + 0.549i)3-s + (0.573 + 1.76i)4-s + (0.937 − 0.680i)5-s + (−0.789 + 0.573i)6-s + (−0.116 − 0.359i)7-s + (−0.447 + 1.37i)8-s + (−0.269 − 0.195i)9-s + 1.95·10-s + (−0.660 + 0.750i)11-s − 1.07·12-s + (−1.52 − 1.10i)13-s + (0.197 − 0.607i)14-s + (0.206 + 0.635i)15-s + (−0.477 + 0.347i)16-s + (0.617 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.0694 - 0.997i$
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: \((2,\ 231,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76470 + 1.64607i\)
\(L(\frac12)\) \(\approx\) \(1.76470 + 1.64607i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 2.48i)T \)
good2 \( 1 + (-1.93 - 1.40i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.09 + 1.52i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (5.48 + 3.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.54 + 1.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.323 - 0.994i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 + (0.187 + 0.577i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.14 - 5.19i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.76 - 8.50i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.68 + 8.27i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 + (-1.00 + 3.10i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.53 + 1.11i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0537 + 0.165i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.58 - 4.78i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.33T + 67T^{2} \)
71 \( 1 + (2.18 - 1.59i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.182 - 0.561i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.93 - 5.76i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.07 - 5.14i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (9.58 + 6.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.52479198209999695933161606525, −12.14738900534197005417811173208, −10.19934800962446046832581985443, −9.800666388359149410125039333038, −8.104348547723287483056875811544, −7.20370829345358660263463881052, −5.85655699414096987888690982779, −5.18651653818261368267183667592, −4.43108838161651007602865742567, −2.80957941502296216669868667512, 2.09664307140790738987860306724, 2.79867750132441243865181102732, 4.47457503547231273357968849532, 5.77053156356872025649091120115, 6.27728051623974454152427226003, 7.76651433266631817888500822588, 9.564886732528583668366910094351, 10.33895500937191667867869773257, 11.31842862709336214714387439788, 12.11628468005586752217530211780

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