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Properties

Label 2-462-77.62-c1-0-3
Degree $2$
Conductor $462$
Sign $-0.138 - 0.990i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (−0.587 + 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.884 + 1.21i)5-s + (−0.809 + 0.587i)6-s + (−2.53 − 0.763i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s − 1.50·10-s + (2.41 + 2.27i)11-s − i·12-s + (2.55 + 1.85i)13-s + (2.10 − 1.60i)14-s + (0.465 + 1.43i)15-s + (−0.809 + 0.587i)16-s + (−3.63 + 2.64i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.395 + 0.544i)5-s + (−0.330 + 0.239i)6-s + (−0.957 − 0.288i)7-s + (0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s − 0.475·10-s + (0.727 + 0.685i)11-s − 0.288i·12-s + (0.707 + 0.514i)13-s + (0.563 − 0.427i)14-s + (0.120 + 0.369i)15-s + (−0.202 + 0.146i)16-s + (−0.882 + 0.641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860424 + 0.989092i\)
\(L(\frac12)\) \(\approx\) \(0.860424 + 0.989092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (2.53 + 0.763i)T \)
11 \( 1 + (-2.41 - 2.27i)T \)
good5 \( 1 + (-0.884 - 1.21i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-2.55 - 1.85i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.63 - 2.64i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.96 - 6.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.51T + 23T^{2} \)
29 \( 1 + (-0.814 + 0.264i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.69 - 5.08i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.85 + 5.72i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.11 + 9.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.65iT - 43T^{2} \)
47 \( 1 + (1.41 + 0.460i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.331 - 0.240i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (5.04 - 1.63i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.80 - 4.21i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (-11.6 + 8.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.34 + 13.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.69 + 10.5i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.08 + 1.51i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.90iT - 89T^{2} \)
97 \( 1 + (-9.52 + 13.1i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.75783311290220849069810900165, −10.39635472306756523584285196371, −9.180086205475710274427987767488, −8.929865957853488376194011586178, −7.52039571702238571507686209750, −6.66548631657313841419020115429, −6.10262792448118750932910443587, −4.43969126204731776350757784479, −3.39720620618936495629496839733, −1.80625771173810404622187962168, 0.940334215784496519407879397366, 2.60816266363221352893296399272, 3.50364349695946626353415310025, 4.91632918680509301992900186765, 6.30178619517535806960487255452, 7.14003561813202176220295915681, 8.612212579051765688872588348352, 9.020401202015587423229620777400, 9.592800039318950615902116206438, 10.89705919497731156764797664391

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