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Properties

Label 2-231-7.4-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.447 - 0.894i$
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  + (0.758 + 1.31i)2-s + (−0.5 + 0.866i)3-s + (−0.150 + 0.259i)4-s + (1.16 + 2.02i)5-s − 1.51·6-s + (−2.28 + 1.33i)7-s + 2.57·8-s + (−0.499 − 0.866i)9-s + (−1.76 + 3.06i)10-s + (0.5 − 0.866i)11-s + (−0.150 − 0.259i)12-s − 1.53·13-s + (−3.48 − 1.99i)14-s − 2.33·15-s + (2.25 + 3.90i)16-s + (0.0583 − 0.100i)17-s + ⋯
L(s)  = 1  + (0.536 + 0.928i)2-s + (−0.288 + 0.499i)3-s + (−0.0750 + 0.129i)4-s + (0.521 + 0.903i)5-s − 0.619·6-s + (−0.863 + 0.503i)7-s + 0.911·8-s + (−0.166 − 0.288i)9-s + (−0.559 + 0.969i)10-s + (0.150 − 0.261i)11-s + (−0.0433 − 0.0750i)12-s − 0.426·13-s + (−0.930 − 0.532i)14-s − 0.602·15-s + (0.563 + 0.976i)16-s + (0.0141 − 0.0244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823578 + 1.33363i\)
\(L(\frac12)\) \(\approx\) \(0.823578 + 1.33363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.28 - 1.33i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.758 - 1.31i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.16 - 2.02i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.53T + 13T^{2} \)
17 \( 1 + (-0.0583 + 0.100i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.80 - 3.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.55 + 6.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.05T + 29T^{2} \)
31 \( 1 + (-2.18 + 3.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.150 - 0.259i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 + 4.18T + 43T^{2} \)
47 \( 1 + (1.15 + 1.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.94 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.47 - 4.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.28 - 3.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.14 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 + (5.21 - 9.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.15 + 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (6.96 + 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−12.65357240178834929240330456985, −11.55442596458975624279135017561, −10.23522610839707926367524218065, −9.998994944015125047632419234112, −8.464075006960242622072524791477, −7.04152226019942325346836589148, −6.24831120072944710353303536177, −5.63654064054955552914214896269, −4.22976978428524002883420212724, −2.67840473864710471943547166712, 1.33102849485020367592514779083, 2.87869320737110597615815433407, 4.31065931407776408285940226025, 5.41786229183012512789774573065, 6.78921289695172486685822829569, 7.78848569429691736254616152996, 9.279904422257295317106010910004, 10.08420239947782887092589088024, 11.16034841761835251744458678804, 12.21021552839171388295714308701

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