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Properties

Label 2-231-11.9-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.929 + 0.369i$
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  + (−0.648 + 1.99i)2-s + (−0.809 + 0.587i)3-s + (−1.94 − 1.41i)4-s + (0.976 + 3.00i)5-s + (−0.648 − 1.99i)6-s + (0.809 + 0.587i)7-s + (0.679 − 0.493i)8-s + (0.309 − 0.951i)9-s − 6.62·10-s + (−0.965 + 3.17i)11-s + 2.40·12-s + (0.657 − 2.02i)13-s + (−1.69 + 1.23i)14-s + (−2.55 − 1.85i)15-s + (−0.939 − 2.89i)16-s + (−1.48 − 4.55i)17-s + ⋯
L(s)  = 1  + (−0.458 + 1.41i)2-s + (−0.467 + 0.339i)3-s + (−0.970 − 0.705i)4-s + (0.436 + 1.34i)5-s + (−0.264 − 0.814i)6-s + (0.305 + 0.222i)7-s + (0.240 − 0.174i)8-s + (0.103 − 0.317i)9-s − 2.09·10-s + (−0.291 + 0.956i)11-s + 0.692·12-s + (0.182 − 0.561i)13-s + (−0.453 + 0.329i)14-s + (−0.659 − 0.479i)15-s + (−0.234 − 0.722i)16-s + (−0.359 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149934 - 0.782694i\)
\(L(\frac12)\) \(\approx\) \(0.149934 - 0.782694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.965 - 3.17i)T \)
good2 \( 1 + (0.648 - 1.99i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.976 - 3.00i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.657 + 2.02i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.48 + 4.55i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.24 - 0.904i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 + (-0.775 - 0.563i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.23 - 6.86i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.94 + 1.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.215 - 0.156i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.28T + 43T^{2} \)
47 \( 1 + (8.35 - 6.06i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.292 - 0.899i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.72 - 5.61i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.64 - 8.13i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + (-4.37 - 13.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.24 + 4.53i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.87 + 8.85i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.20 + 16.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-2.19 + 6.74i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.85424671815464609601354678257, −11.52572677148230759265228376698, −10.60626863931953929073482181905, −9.752639419708387166597293373747, −8.730622924880327937202782862505, −7.34793335572357137380942292262, −6.89178198174946815749231113994, −5.80657258813701380976155641965, −4.86330352092261383880909911642, −2.79474261131064137530859781915, 0.832891437684013252612664451843, 2.03873145263025525450799693367, 3.91321379163458574482648273242, 5.18350748046861704389517369875, 6.40572270229380919495623775096, 8.230340266157393320369107715065, 8.875583908972978672977517272016, 9.837509784837174499415959287843, 11.00898644607565888635125774163, 11.39872444974633973719161056609

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