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Properties

Label 2-231-7.2-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.0932 + 0.995i$
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  + (−1.29 + 2.24i)2-s + (0.5 + 0.866i)3-s + (−2.36 − 4.09i)4-s + (−1.09 + 1.89i)5-s − 2.59·6-s + (−2.61 − 0.413i)7-s + 7.08·8-s + (−0.499 + 0.866i)9-s + (−2.84 − 4.92i)10-s + (−0.5 − 0.866i)11-s + (2.36 − 4.09i)12-s − 2.95·13-s + (4.31 − 5.33i)14-s − 2.19·15-s + (−4.46 + 7.72i)16-s + (−1.29 − 2.24i)17-s + ⋯
L(s)  = 1  + (−0.917 + 1.58i)2-s + (0.288 + 0.499i)3-s + (−1.18 − 2.04i)4-s + (−0.490 + 0.848i)5-s − 1.05·6-s + (−0.987 − 0.156i)7-s + 2.50·8-s + (−0.166 + 0.288i)9-s + (−0.899 − 1.55i)10-s + (−0.150 − 0.261i)11-s + (0.682 − 1.18i)12-s − 0.818·13-s + (1.15 − 1.42i)14-s − 0.565·15-s + (−1.11 + 1.93i)16-s + (−0.314 − 0.544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0932 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0932 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.0932 + 0.995i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.0932 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181363 - 0.199153i\)
\(L(\frac12)\) \(\approx\) \(0.181363 - 0.199153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.413i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.29 - 2.24i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.09 - 1.89i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 + (1.29 + 2.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 + 2.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.46 - 7.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.48T + 29T^{2} \)
31 \( 1 + (-0.865 - 1.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.19 + 3.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.68T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + (5.55 - 9.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.58 - 7.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.46 + 7.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.73 - 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.62 - 2.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.994T + 71T^{2} \)
73 \( 1 + (0.147 + 0.255i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.53 + 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.26T + 83T^{2} \)
89 \( 1 + (-1.30 + 2.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.51T + 97T^{2} \)
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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

−13.43282627843207345545350358201, −11.69146759551105142434615136928, −10.41210541046388988415811996658, −9.760316754171655244292703313912, −8.970417642896215041508778375116, −7.73959020040597972427653822105, −7.11483406429480443170111230519, −6.15714113498764549512025125827, −4.88990762535207340058718319067, −3.23111926566222503422773079308, 0.27680747400590555259169607297, 2.07739223728395573809985178677, 3.33911796372841935173982936034, 4.60145515814100237089663011469, 6.69744334138222246728278895052, 8.204804619151186449899252666352, 8.537235262602306700562274101694, 9.798162334609571884789850381072, 10.25232751114841272551447777666, 11.77012936426062663833679013244

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