Properties

Label 2-231-7.4-c1-0-8
Degree $2$
Conductor $231$
Sign $0.919 + 0.394i$
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  + (0.276 + 0.478i)2-s + (−0.5 + 0.866i)3-s + (0.847 − 1.46i)4-s + (−0.795 − 1.37i)5-s − 0.552·6-s + (0.886 − 2.49i)7-s + 2.04·8-s + (−0.499 − 0.866i)9-s + (0.439 − 0.760i)10-s + (0.5 − 0.866i)11-s + (0.847 + 1.46i)12-s + 2.87·13-s + (1.43 − 0.264i)14-s + 1.59·15-s + (−1.13 − 1.95i)16-s + (−2.41 + 4.18i)17-s + ⋯
L(s)  = 1  + (0.195 + 0.338i)2-s + (−0.288 + 0.499i)3-s + (0.423 − 0.733i)4-s + (−0.355 − 0.615i)5-s − 0.225·6-s + (0.335 − 0.942i)7-s + 0.721·8-s + (−0.166 − 0.288i)9-s + (0.138 − 0.240i)10-s + (0.150 − 0.261i)11-s + (0.244 + 0.423i)12-s + 0.798·13-s + (0.384 − 0.0706i)14-s + 0.410·15-s + (−0.282 − 0.489i)16-s + (−0.586 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.919 + 0.394i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.919 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32803 - 0.272772i\)
\(L(\frac12)\) \(\approx\) \(1.32803 - 0.272772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.886 + 2.49i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.276 - 0.478i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.795 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 + (2.41 - 4.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.572 - 0.992i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.82 - 3.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.325T + 29T^{2} \)
31 \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.847 + 1.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.05T + 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 + (0.152 + 0.264i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.85 - 4.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.93 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.886 + 1.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.63 - 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 + (-5.86 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.35 - 7.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.40T + 83T^{2} \)
89 \( 1 + (-2.87 - 4.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−11.99658263047506728482618121274, −10.83363965050278833422016042324, −10.61891236319438271090376436371, −9.238519670720707235888863775354, −8.149600442754550098440076774280, −6.93964968946083329810763053426, −5.90060907194582771522793469271, −4.80341511601323469370084678060, −3.82330415772029047684426536274, −1.28258670367271263540369127726, 2.14386767711641950425743656172, 3.29908387171146754628571430667, 4.83914330511218405621535077545, 6.33752613269677379715130109822, 7.21316077415277877595608713118, 8.165287255080052622012695973571, 9.230628080110869304784679199463, 10.96031542476189057089046745808, 11.30701123102849467700965134262, 12.15796735576492853314423815212

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