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Properties

Label 2-462-231.65-c1-0-11
Degree $2$
Conductor $462$
Sign $0.764 - 0.645i$
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  + (−0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.13 + 0.656i)5-s + 1.73i·6-s + (1.13 + 2.38i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (−1.13 + 0.656i)10-s + (0.637 + 3.25i)11-s + (−1.49 − 0.866i)12-s − 2.62i·13-s + (−2.63 − 0.209i)14-s + 2.27·15-s + (−0.5 + 0.866i)16-s + (1.63 + 2.83i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.508 + 0.293i)5-s + 0.707i·6-s + (0.429 + 0.902i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.359 + 0.207i)10-s + (0.192 + 0.981i)11-s + (−0.433 − 0.250i)12-s − 0.728i·13-s + (−0.704 − 0.0559i)14-s + 0.587·15-s + (−0.125 + 0.216i)16-s + (0.397 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.764 - 0.645i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.764 - 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63528 + 0.598022i\)
\(L(\frac12)\) \(\approx\) \(1.63528 + 0.598022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-1.13 - 2.38i)T \)
11 \( 1 + (-0.637 - 3.25i)T \)
good5 \( 1 + (-1.13 - 0.656i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 + (-1.63 - 2.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.63 + 3.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.91 - 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-5.13 - 8.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.63 + 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.54T + 41T^{2} \)
43 \( 1 + 9.97iT - 43T^{2} \)
47 \( 1 + (1.91 + 1.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.6 - 6.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.27 - 7.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.97iT - 71T^{2} \)
73 \( 1 + (4.54 - 2.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.86 + 1.07i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (2.27 + 1.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−10.92865690886929139733546260707, −9.989038835035449758539431139631, −9.104308979973973397981646580586, −8.469895240369365119027993413191, −7.55766195598256490420651133990, −6.66203133104305873301886822119, −5.75226885464158588717406942495, −4.48261076331967985419411705959, −2.79682407681572668185502401679, −1.70333434113284914696913424372, 1.38136662430918701728729888055, 2.76535669780637322480292955984, 3.98383083553465758531094576365, 4.77353454852939250129448873877, 6.34907847785201499892056574163, 7.72733367602777483952212944839, 8.363565033654378812716695891522, 9.351537379581503466380526350490, 9.883403245060343013614574780709, 10.92503456649214162348524886859

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