Properties

Label 2-231-11.3-c1-0-3
Degree $2$
Conductor $231$
Sign $-0.440 - 0.897i$
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  + (−0.705 + 0.512i)2-s + (0.309 + 0.951i)3-s + (−0.383 + 1.17i)4-s + (3.28 + 2.38i)5-s + (−0.705 − 0.512i)6-s + (−0.309 + 0.951i)7-s + (−0.872 − 2.68i)8-s + (−0.809 + 0.587i)9-s − 3.54·10-s + (−2.96 − 1.48i)11-s − 1.24·12-s + (4.92 − 3.57i)13-s + (−0.269 − 0.828i)14-s + (−1.25 + 3.86i)15-s + (−0.0153 − 0.0111i)16-s + (−1.37 − 1.00i)17-s + ⋯
L(s)  = 1  + (−0.498 + 0.362i)2-s + (0.178 + 0.549i)3-s + (−0.191 + 0.589i)4-s + (1.47 + 1.06i)5-s + (−0.287 − 0.209i)6-s + (−0.116 + 0.359i)7-s + (−0.308 − 0.949i)8-s + (−0.269 + 0.195i)9-s − 1.12·10-s + (−0.894 − 0.446i)11-s − 0.358·12-s + (1.36 − 0.992i)13-s + (−0.0719 − 0.221i)14-s + (−0.324 + 0.998i)15-s + (−0.00383 − 0.00278i)16-s + (−0.333 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.440 - 0.897i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598918 + 0.960476i\)
\(L(\frac12)\) \(\approx\) \(0.598918 + 0.960476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (2.96 + 1.48i)T \)
good2 \( 1 + (0.705 - 0.512i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-3.28 - 2.38i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.92 + 3.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.37 + 1.00i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.68 - 5.19i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.39T + 23T^{2} \)
29 \( 1 + (-1.46 + 4.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.52 + 2.56i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.753 + 2.32i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.992 + 3.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.127T + 43T^{2} \)
47 \( 1 + (-2.61 - 8.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.81 + 2.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.43 + 4.40i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.29 + 3.12i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + (1.79 + 1.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.72 - 5.30i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.07 - 3.68i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.103 - 0.0750i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.12T + 89T^{2} \)
97 \( 1 + (-2.10 + 1.52i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.73017441654917654121353630173, −11.23340105642225035359659481539, −10.20960931522998654413375552637, −9.770751279085897476189312849949, −8.562390340261654210187603196684, −7.77560236221261264875873557322, −6.25020967997906552649209792705, −5.68922988845287931831337289327, −3.63409622504588684287374530609, −2.59029320353938008005655034576, 1.21863370719623559506071544362, 2.24488964203738058932447451798, 4.67790924537485488534269721373, 5.72579747906606545886666812640, 6.65097802868231613376671496246, 8.405671216667906489765218471553, 8.997653061544666893670398608543, 9.877400409491674110206513168239, 10.66291742198238712117452991555, 11.87389465609368162128052771681

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