Properties

Label 2-231-231.32-c1-0-9
Degree $2$
Conductor $231$
Sign $0.294 - 0.955i$
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  + (0.939 + 1.62i)2-s + (−1.69 − 0.370i)3-s + (−0.765 + 1.32i)4-s + (2.63 − 1.52i)5-s + (−0.986 − 3.10i)6-s + (0.738 + 2.54i)7-s + 0.882·8-s + (2.72 + 1.25i)9-s + (4.95 + 2.86i)10-s + (−3.23 + 0.726i)11-s + (1.78 − 1.95i)12-s + 1.24i·13-s + (−3.44 + 3.58i)14-s + (−5.02 + 1.59i)15-s + (2.35 + 4.08i)16-s + (2.83 − 4.90i)17-s + ⋯
L(s)  = 1  + (0.664 + 1.15i)2-s + (−0.976 − 0.213i)3-s + (−0.382 + 0.662i)4-s + (1.17 − 0.680i)5-s + (−0.402 − 1.26i)6-s + (0.279 + 0.960i)7-s + 0.311·8-s + (0.908 + 0.417i)9-s + (1.56 + 0.904i)10-s + (−0.975 + 0.219i)11-s + (0.515 − 0.565i)12-s + 0.345i·13-s + (−0.919 + 0.959i)14-s + (−1.29 + 0.412i)15-s + (0.589 + 1.02i)16-s + (0.687 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.294 - 0.955i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27690 + 0.942794i\)
\(L(\frac12)\) \(\approx\) \(1.27690 + 0.942794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.370i)T \)
7 \( 1 + (-0.738 - 2.54i)T \)
11 \( 1 + (3.23 - 0.726i)T \)
good2 \( 1 + (-0.939 - 1.62i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.63 + 1.52i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.24iT - 13T^{2} \)
17 \( 1 + (-2.83 + 4.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.532 - 0.307i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.41 - 2.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.47T + 29T^{2} \)
31 \( 1 + (-3.18 + 5.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.15 + 5.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 2.31iT - 43T^{2} \)
47 \( 1 + (-9.35 + 5.40i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.58 + 1.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.38 + 3.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.24 - 4.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.31 + 4.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + (-1.01 - 0.586i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.34 + 1.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + (1.19 - 0.690i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.19T + 97T^{2} \)
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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

−12.56200885145438675508348576257, −11.77385486712962170132597730704, −10.39118672816424364623847717131, −9.503599224875873953796371370134, −8.125641195257131488057456028566, −7.06655544846954692406362260441, −5.82594629354483138043991617454, −5.49560431129547543242135669567, −4.68914529570910334435840222825, −1.94903533539781190651640418736, 1.58001912040725462623554704675, 3.16374713226737198732059347010, 4.49477984864884832054030261484, 5.55696940865807641821211552584, 6.59348421103178133712463274573, 7.916559011627581695678480137357, 10.01973384621985562971517475601, 10.42989628512808198075088392321, 10.76675357421682420097475636068, 12.01241249499268961881971130816

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