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Properties

Label 2-462-7.4-c1-0-11
Degree $2$
Conductor $462$
Sign $-0.999 - 0.00445i$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−2.20 − 3.82i)5-s − 0.999·6-s + (2.20 − 1.45i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−2.20 + 3.82i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (−2.36 − 1.18i)14-s − 4.41·15-s + (−0.5 − 0.866i)16-s + (−1.18 + 2.05i)17-s + (−0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.988 − 1.71i)5-s − 0.408·6-s + (0.835 − 0.550i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.698 + 1.21i)10-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s + (−0.632 − 0.316i)14-s − 1.14·15-s + (−0.125 − 0.216i)16-s + (−0.288 + 0.499i)17-s + (−0.117 + 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.00445i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.999 - 0.00445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00216180 + 0.969975i\)
\(L(\frac12)\) \(\approx\) \(0.00216180 + 0.969975i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.20 + 1.45i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (2.20 + 3.82i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (1.18 - 2.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.84 - 3.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.52T + 29T^{2} \)
31 \( 1 + (-1.68 + 2.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.84 + 6.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 3.06T + 43T^{2} \)
47 \( 1 + (-2.05 - 3.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.26 + 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.89 + 3.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.07 - 8.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.934T + 71T^{2} \)
73 \( 1 + (-3.73 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.209 - 0.362i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.14T + 83T^{2} \)
89 \( 1 + (-6.41 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.85722330216474038869847553121, −9.456160178338210350934241378805, −8.722490921205173670786866929647, −7.966771311112874951398321419733, −7.52008021556158780418060953434, −5.68696509750204597620133698352, −4.42638206651835522727959198060, −3.81247630022705073193174997289, −1.80893258573791624009973629299, −0.67753141026476462879431093064, 2.43674965623233933892798906181, 3.65099212679395032654023090375, 4.78985201766872580628018039073, 6.04882091979215955060644518988, 7.23449214554673464668369159857, 7.65122533038516063087853567115, 8.694046900555677856612644285388, 9.665475994265718443868269589223, 10.64064910430910429989399784472, 11.33258768424340081147725804540

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