Properties

Label 2-462-77.16-c1-0-10
Degree $2$
Conductor $462$
Sign $0.916 + 0.399i$
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  + (0.978 + 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (1.00 + 1.12i)5-s + (0.309 − 0.951i)6-s + (0.191 − 2.63i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (0.753 + 1.30i)10-s + (1.15 − 3.10i)11-s + (0.5 − 0.866i)12-s + (1.02 + 3.16i)13-s + (0.735 − 2.54i)14-s + (1.21 − 0.886i)15-s + (0.669 + 0.743i)16-s + (0.990 − 0.210i)17-s + ⋯
L(s)  = 1  + (0.691 + 0.147i)2-s + (0.0603 − 0.574i)3-s + (0.456 + 0.203i)4-s + (0.451 + 0.501i)5-s + (0.126 − 0.388i)6-s + (0.0722 − 0.997i)7-s + (0.286 + 0.207i)8-s + (−0.326 − 0.0693i)9-s + (0.238 + 0.412i)10-s + (0.348 − 0.937i)11-s + (0.144 − 0.250i)12-s + (0.285 + 0.878i)13-s + (0.196 − 0.679i)14-s + (0.314 − 0.228i)15-s + (0.167 + 0.185i)16-s + (0.240 − 0.0510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29061 - 0.477204i\)
\(L(\frac12)\) \(\approx\) \(2.29061 - 0.477204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (-0.191 + 2.63i)T \)
11 \( 1 + (-1.15 + 3.10i)T \)
good5 \( 1 + (-1.00 - 1.12i)T + (-0.522 + 4.97i)T^{2} \)
13 \( 1 + (-1.02 - 3.16i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.990 + 0.210i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-3.90 + 1.73i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (2.00 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.79 - 5.66i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.09 + 2.32i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-0.333 - 3.17i)T + (-36.1 + 7.69i)T^{2} \)
41 \( 1 + (-3.91 - 2.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.86T + 43T^{2} \)
47 \( 1 + (-2.24 + 0.999i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (0.537 - 0.596i)T + (-5.54 - 52.7i)T^{2} \)
59 \( 1 + (-3.83 - 1.70i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (3.49 + 3.88i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (0.962 + 1.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.87 - 11.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.88 + 2.17i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (13.8 + 2.94i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (3.24 - 9.97i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.34 + 4.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.904 - 2.78i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.30781422704717610159110748814, −10.25897341727582180570055582034, −9.194770176976827717534009415351, −7.978792732634712197454392503412, −7.08265870980589278355238087098, −6.41215518409462132004264246033, −5.43760119015377402300417789221, −4.03008288651536752106090367141, −3.05058850066300145012747153978, −1.46947744427639275168351223061, 1.85799889849582669508447452639, 3.16927111664543717902046214997, 4.40519270223654015713253783473, 5.43329291328371739383105085784, 5.93537889183351753941578348129, 7.42101714929185884491756489565, 8.542949112728333266068707126773, 9.510703016241703884666782784188, 10.10387179980661998364940763404, 11.25958121932394395357887093550

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