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Properties

Label 2-231-33.32-c1-0-20
Degree $2$
Conductor $231$
Sign $0.603 + 0.797i$
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  + 0.858·2-s + (1.60 − 0.639i)3-s − 1.26·4-s − 2.44i·5-s + (1.38 − 0.549i)6-s − i·7-s − 2.80·8-s + (2.18 − 2.05i)9-s − 2.10i·10-s + (3.19 − 0.883i)11-s + (−2.03 + 0.807i)12-s + 5.20i·13-s − 0.858i·14-s + (−1.56 − 3.93i)15-s + 0.121·16-s − 0.151·17-s + ⋯
L(s)  = 1  + 0.607·2-s + (0.929 − 0.369i)3-s − 0.631·4-s − 1.09i·5-s + (0.564 − 0.224i)6-s − 0.377i·7-s − 0.990·8-s + (0.727 − 0.686i)9-s − 0.664i·10-s + (0.963 − 0.266i)11-s + (−0.586 + 0.233i)12-s + 1.44i·13-s − 0.229i·14-s + (−0.404 − 1.01i)15-s + 0.0303·16-s − 0.0367·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65365 - 0.822316i\)
\(L(\frac12)\) \(\approx\) \(1.65365 - 0.822316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 + 0.639i)T \)
7 \( 1 + iT \)
11 \( 1 + (-3.19 + 0.883i)T \)
good2 \( 1 - 0.858T + 2T^{2} \)
5 \( 1 + 2.44iT - 5T^{2} \)
13 \( 1 - 5.20iT - 13T^{2} \)
17 \( 1 + 0.151T + 17T^{2} \)
19 \( 1 - 3.38iT - 19T^{2} \)
23 \( 1 - 5.75iT - 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 + 9.44iT - 43T^{2} \)
47 \( 1 - 0.242iT - 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 2.22iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 1.94T + 67T^{2} \)
71 \( 1 - 7.57iT - 71T^{2} \)
73 \( 1 - 3.49iT - 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 2.58iT - 89T^{2} \)
97 \( 1 + 5.06T + 97T^{2} \)
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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

−12.39738355633243102866800789718, −11.50629776260725299457343243803, −9.519425237743599847401828732661, −9.211966074056886728645930205968, −8.314138630850710111544784359455, −7.06021984364004914707281723235, −5.71169078292474831595954492380, −4.29249668765070173004287349577, −3.70194977521935449619234535094, −1.51368302683149333294965803185, 2.69917428327419456465923326553, 3.58202431578965731350600973800, 4.76688816932607302676210379583, 6.11070122866744036849735278425, 7.37701438994599390114488819527, 8.545382281952166466146980733902, 9.399814466149255554376242050175, 10.30176830940011811278197090316, 11.35876980998521615683682552351, 12.74888188361892868793210315468

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