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Properties

Label 2-231-231.65-c1-0-1
Degree $2$
Conductor $231$
Sign $-0.997 + 0.0699i$
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  + (−0.794 + 1.37i)2-s + (1.04 + 1.37i)3-s + (−0.262 − 0.454i)4-s + (−1.14 − 0.662i)5-s + (−2.73 + 0.348i)6-s + (−2.48 − 0.904i)7-s − 2.34·8-s + (−0.798 + 2.89i)9-s + (1.82 − 1.05i)10-s + (−2.42 + 2.26i)11-s + (0.351 − 0.839i)12-s + 3.14i·13-s + (3.22 − 2.70i)14-s + (−0.290 − 2.27i)15-s + (2.38 − 4.13i)16-s + (1.28 + 2.22i)17-s + ⋯
L(s)  = 1  + (−0.561 + 0.973i)2-s + (0.605 + 0.795i)3-s + (−0.131 − 0.227i)4-s + (−0.513 − 0.296i)5-s + (−1.11 + 0.142i)6-s + (−0.939 − 0.341i)7-s − 0.828·8-s + (−0.266 + 0.963i)9-s + (0.576 − 0.333i)10-s + (−0.730 + 0.683i)11-s + (0.101 − 0.242i)12-s + 0.870i·13-s + (0.860 − 0.722i)14-s + (−0.0751 − 0.588i)15-s + (0.596 − 1.03i)16-s + (0.312 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.997 + 0.0699i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.997 + 0.0699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0268547 - 0.767071i\)
\(L(\frac12)\) \(\approx\) \(0.0268547 - 0.767071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.04 - 1.37i)T \)
7 \( 1 + (2.48 + 0.904i)T \)
11 \( 1 + (2.42 - 2.26i)T \)
good2 \( 1 + (0.794 - 1.37i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.14 + 0.662i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.14iT - 13T^{2} \)
17 \( 1 + (-1.28 - 2.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.54 - 2.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.23 - 3.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.92T + 29T^{2} \)
31 \( 1 + (0.543 + 0.940i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.551 - 0.955i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.61T + 41T^{2} \)
43 \( 1 + 5.86iT - 43T^{2} \)
47 \( 1 + (1.18 + 0.685i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.25 + 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.38 - 5.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.35 - 4.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.67 - 8.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (11.3 - 6.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.91 + 3.41i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + (-11.1 - 6.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.94T + 97T^{2} \)
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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

−12.74389225110141938770408972112, −11.73113653430707100727729297180, −10.32195997535443776177775018265, −9.554904090980488158049383586171, −8.734441133910051333644689058298, −7.74864430633023731198608697996, −7.01832645217500223692773730301, −5.59260142251025181636463284442, −4.18933352742117099090311158285, −2.98065594193183007987524991523, 0.69433555691739032041583193655, 2.85307557852789721402452794336, 3.13281114297547282820515816266, 5.64257272881048193741677155524, 6.81856210849282139441473555302, 7.926632656825579313445739700589, 8.925898101364381193245734228556, 9.736345501814810740322203385338, 10.82054701938049763486552150092, 11.65263472308315985958556927160

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