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Properties

Label 2-462-77.41-c1-0-8
Degree $2$
Conductor $462$
Sign $0.742 - 0.669i$
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 + (0.531 − 0.731i)5-s + (−0.809 − 0.587i)6-s + (1.75 − 1.97i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + 0.904·10-s + (2.92 + 1.57i)11-s − i·12-s + (1.17 − 0.856i)13-s + (2.63 + 0.261i)14-s + (−0.279 + 0.860i)15-s + (−0.809 − 0.587i)16-s + (2.41 + 1.75i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.237 − 0.327i)5-s + (−0.330 − 0.239i)6-s + (0.664 − 0.747i)7-s + (−0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s + 0.285·10-s + (0.880 + 0.474i)11-s − 0.288i·12-s + (0.326 − 0.237i)13-s + (0.703 + 0.0697i)14-s + (−0.0721 + 0.222i)15-s + (−0.202 − 0.146i)16-s + (0.584 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.742 - 0.669i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57487 + 0.604925i\)
\(L(\frac12)\) \(\approx\) \(1.57487 + 0.604925i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 1.97i)T \)
11 \( 1 + (-2.92 - 1.57i)T \)
good5 \( 1 + (-0.531 + 0.731i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.17 + 0.856i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.41 - 1.75i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0587 + 0.180i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.71T + 23T^{2} \)
29 \( 1 + (-1.14 - 0.372i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.65 + 2.27i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.615 - 1.89i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.71 - 8.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.16iT - 43T^{2} \)
47 \( 1 + (11.9 - 3.88i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-11.6 + 8.46i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.25 - 2.35i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.85 + 3.52i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 + (7.86 + 5.71i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.43 + 4.42i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.69 + 11.9i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.97 - 5.06i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.06iT - 89T^{2} \)
97 \( 1 + (2.47 + 3.39i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.27404008539540019147378073246, −10.30370641667303342488899862649, −9.347970336479017232697761292523, −8.298090335896758425442903762519, −7.32386242268300970601079829130, −6.46072667573857182532610965012, −5.40310756777640191460718080295, −4.56292643141890790035158024722, −3.59188840529024045548244939384, −1.37976270987666553123638810049, 1.35279818453565022152563051031, 2.73066357910665352632432595974, 4.11641477292183835076761076313, 5.27743561767544834967352308523, 6.04811560468937994346966180510, 7.01689838844681954077491059041, 8.400027304780669436051834706307, 9.248251794081697542342169482271, 10.31601034285411946673256921776, 11.16049305248608284156778176657

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