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Properties

Label 2-231-7.4-c1-0-5
Degree $2$
Conductor $231$
Sign $-0.532 - 0.846i$
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  + (1.24 + 2.15i)2-s + (0.5 − 0.866i)3-s + (−2.10 + 3.64i)4-s + (0.440 + 0.762i)5-s + 2.49·6-s + (−2.14 + 1.54i)7-s − 5.52·8-s + (−0.499 − 0.866i)9-s + (−1.09 + 1.90i)10-s + (−0.5 + 0.866i)11-s + (2.10 + 3.64i)12-s + 7.12·13-s + (−6.01 − 2.70i)14-s + 0.880·15-s + (−2.66 − 4.61i)16-s + (1.24 − 2.15i)17-s + ⋯
L(s)  = 1  + (0.881 + 1.52i)2-s + (0.288 − 0.499i)3-s + (−1.05 + 1.82i)4-s + (0.196 + 0.341i)5-s + 1.01·6-s + (−0.811 + 0.584i)7-s − 1.95·8-s + (−0.166 − 0.288i)9-s + (−0.347 + 0.601i)10-s + (−0.150 + 0.261i)11-s + (0.608 + 1.05i)12-s + 1.97·13-s + (−1.60 − 0.722i)14-s + 0.227·15-s + (−0.666 − 1.15i)16-s + (0.302 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.532 - 0.846i$
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: \((2,\ 231,\ (\ :1/2),\ -0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.918884 + 1.66358i\)
\(L(\frac12)\) \(\approx\) \(0.918884 + 1.66358i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 1.54i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.24 - 2.15i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.440 - 0.762i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 7.12T + 13T^{2} \)
17 \( 1 + (-1.24 + 2.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.31 + 2.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.66 + 4.61i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.773T + 29T^{2} \)
31 \( 1 + (-0.607 + 1.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.43 + 7.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.19T + 41T^{2} \)
43 \( 1 - 3.22T + 43T^{2} \)
47 \( 1 + (-4.87 - 8.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.46 - 7.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.66 - 4.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.54 - 7.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.63 - 9.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 + (-6.38 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.47 - 6.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.20T + 83T^{2} \)
89 \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.29T + 97T^{2} \)
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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

−12.87770298515721160478664977756, −12.13788784041561491972919037856, −10.62011191490644776078099737260, −9.034598717822460437251969317719, −8.389242974042674736837135806537, −7.18668150951093719511335377400, −6.33467660539963283736138884945, −5.77585293460121336607540616613, −4.18329560277037879040960237098, −2.90730816725107387492205161598, 1.48008530645475742953934655230, 3.41080017339644472557022223623, 3.77902549510911065146059293834, 5.23922588403137150451475079572, 6.29998373133151050351495628543, 8.325091688380609568286789258275, 9.385413304152735178574181335689, 10.29165481569193263352415961710, 10.85793568830488340458282777107, 11.86370337312799458363225324181

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