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Properties

Label 2-231-77.10-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.999 + 0.0277i$
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  + (−0.360 + 0.208i)2-s + (−0.866 − 0.5i)3-s + (−0.913 + 1.58i)4-s + (0.414 − 0.239i)5-s + 0.416·6-s + (−2.50 − 0.863i)7-s − 1.59i·8-s + (0.499 + 0.866i)9-s + (−0.0995 + 0.172i)10-s + (−2.41 + 2.27i)11-s + (1.58 − 0.913i)12-s − 5.34·13-s + (1.08 − 0.209i)14-s − 0.478·15-s + (−1.49 − 2.58i)16-s + (−0.123 + 0.213i)17-s + ⋯
L(s)  = 1  + (−0.255 + 0.147i)2-s + (−0.499 − 0.288i)3-s + (−0.456 + 0.790i)4-s + (0.185 − 0.106i)5-s + 0.170·6-s + (−0.945 − 0.326i)7-s − 0.563i·8-s + (0.166 + 0.288i)9-s + (−0.0314 + 0.0545i)10-s + (−0.726 + 0.686i)11-s + (0.456 − 0.263i)12-s − 1.48·13-s + (0.289 − 0.0559i)14-s − 0.123·15-s + (−0.373 − 0.647i)16-s + (−0.0298 + 0.0517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.999 + 0.0277i$
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: \((2,\ 231,\ (\ :1/2),\ -0.999 + 0.0277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00140121 - 0.101080i\)
\(L(\frac12)\) \(\approx\) \(0.00140121 - 0.101080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.863i)T \)
11 \( 1 + (2.41 - 2.27i)T \)
good2 \( 1 + (0.360 - 0.208i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.414 + 0.239i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.34T + 13T^{2} \)
17 \( 1 + (0.123 - 0.213i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.02 + 3.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.540 + 0.936i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.25iT - 29T^{2} \)
31 \( 1 + (1.00 + 0.580i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.23 + 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.10T + 41T^{2} \)
43 \( 1 + 2.69iT - 43T^{2} \)
47 \( 1 + (-6.81 + 3.93i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.242 + 0.420i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.959 + 0.553i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.66 - 4.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.52 - 2.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (3.94 - 6.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.3 - 7.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + (-11.9 + 6.92i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.20iT - 97T^{2} \)
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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

−12.80405681352991245786469371455, −12.02043647362298843331144334373, −10.60581072866027737382788080908, −9.729123096756549090575593191513, −8.871239174581491878260176463087, −7.32557406310664003785577561663, −7.11957898727419771301406941362, −5.43632433683827987936245663979, −4.28465978808539206438797467098, −2.68827607578542369831551896427, 0.089462001657176143203016438982, 2.53100358308475758484728296235, 4.36837805018961672780748193181, 5.62586177238445253794301444450, 6.23255505569778546391538351063, 7.81211467000199203113035781922, 9.121031779603909646672155098908, 9.980293444278516443056742170685, 10.39497720717372774360661328641, 11.63511235168770952013208312275

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