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Properties

Label 2-231-7.5-c2-0-14
Degree $2$
Conductor $231$
Sign $0.829 + 0.558i$
Analytic cond. $6.29429$
Root an. cond. $2.50884$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.482 − 0.835i)2-s + (1.5 − 0.866i)3-s + (1.53 + 2.65i)4-s + (−3.16 − 1.82i)5-s − 1.67i·6-s + (6.98 − 0.404i)7-s + 6.82·8-s + (1.5 − 2.59i)9-s + (−3.05 + 1.76i)10-s + (−1.65 − 2.87i)11-s + (4.60 + 2.65i)12-s + 0.490i·13-s + (3.03 − 6.03i)14-s − 6.33·15-s + (−2.84 + 4.93i)16-s + (18.6 − 10.7i)17-s + ⋯
L(s)  = 1  + (0.241 − 0.417i)2-s + (0.5 − 0.288i)3-s + (0.383 + 0.664i)4-s + (−0.633 − 0.365i)5-s − 0.278i·6-s + (0.998 − 0.0577i)7-s + 0.852·8-s + (0.166 − 0.288i)9-s + (−0.305 + 0.176i)10-s + (−0.150 − 0.261i)11-s + (0.383 + 0.221i)12-s + 0.0377i·13-s + (0.216 − 0.431i)14-s − 0.422·15-s + (−0.177 + 0.308i)16-s + (1.09 − 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(6.29429\)
Root analytic conductor: \(2.50884\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.24945 - 0.686662i\)
\(L(\frac12)\) \(\approx\) \(2.24945 - 0.686662i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-6.98 + 0.404i)T \)
11 \( 1 + (1.65 + 2.87i)T \)
good2 \( 1 + (-0.482 + 0.835i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (3.16 + 1.82i)T + (12.5 + 21.6i)T^{2} \)
13 \( 1 - 0.490iT - 169T^{2} \)
17 \( 1 + (-18.6 + 10.7i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-27.1 - 15.6i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.70 + 6.42i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 56.8T + 841T^{2} \)
31 \( 1 + (0.924 - 0.534i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (8.73 - 15.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 14.7iT - 1.68e3T^{2} \)
43 \( 1 + 48.6T + 1.84e3T^{2} \)
47 \( 1 + (-23.7 - 13.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.498 - 0.862i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (55.1 - 31.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (46.2 + 26.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.9 - 69.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 78.8T + 5.04e3T^{2} \)
73 \( 1 + (83.2 - 48.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (70.5 - 122. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 20.7iT - 6.88e3T^{2} \)
89 \( 1 + (19.2 + 11.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 68.2iT - 9.40e3T^{2} \)
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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

−11.84815001569052100386818134885, −11.35181895618096011006426278363, −10.05065062828594696263329722085, −8.674659857062173212991970410545, −7.75921905936041997645585790405, −7.38352991108473195589369874399, −5.43995474776814453879776726965, −4.11198191311289238643779759650, −3.07674717993488424741483268814, −1.49567875357790372361288019006, 1.66308191044364182284972391905, 3.41903000491278892425899312494, 4.82903031522273659808452011100, 5.71617027816159728330746531145, 7.44351833566774888713725609416, 7.62453568490106646746111052487, 9.146796269514553624041066773621, 10.20698479597759240644564534218, 11.13676407360645637995358847046, 11.78655286028349279551803455413

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