Properties

Label 2-231-11.4-c1-0-10
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: $\chi_{231} (169, \cdot )$
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

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

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