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Properties

Label 2-462-11.9-c1-0-6
Degree $2$
Conductor $462$
Sign $0.959 + 0.282i$
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  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.530 − 1.63i)5-s + (0.309 + 0.951i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 1.71·10-s + (0.969 + 3.17i)11-s − 0.999·12-s + (0.931 − 2.86i)13-s + (−0.809 + 0.587i)14-s + (−1.38 − 1.00i)15-s + (0.309 + 0.951i)16-s + (−1.5 − 4.61i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.237 − 0.730i)5-s + (0.126 + 0.388i)6-s + (0.305 + 0.222i)7-s + (0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + 0.542·10-s + (0.292 + 0.956i)11-s − 0.288·12-s + (0.258 − 0.795i)13-s + (−0.216 + 0.157i)14-s + (−0.358 − 0.260i)15-s + (0.0772 + 0.237i)16-s + (−0.363 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40863 - 0.202772i\)
\(L(\frac12)\) \(\approx\) \(1.40863 - 0.202772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.969 - 3.17i)T \)
good5 \( 1 + (0.530 + 1.63i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.931 + 2.86i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.5 + 4.61i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.36 + 3.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.18T + 23T^{2} \)
29 \( 1 + (6.56 + 4.76i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.938 + 2.88i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.62 - 5.53i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.63 - 2.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.30T + 43T^{2} \)
47 \( 1 + (8.18 - 5.94i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.91 - 12.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.56 + 3.31i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.377 - 1.16i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 5.69T + 67T^{2} \)
71 \( 1 + (1.17 + 3.61i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.945 - 0.687i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.86 - 14.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.14 - 3.51i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 18.7T + 89T^{2} \)
97 \( 1 + (-0.957 + 2.94i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.09074685015535447928052070791, −9.537141433053488435001135608300, −9.274360312745590429470033769805, −8.129634109997428742370341456501, −7.51496574989570833620540136210, −6.57181897889219861728320034514, −5.20831425019770530905558678477, −4.51862737944350227945583482473, −2.84374612069673425259111416689, −1.05366939173866388662281702616, 1.61854282608062151655352988114, 3.22450765177209399433030389518, 3.78967041931947994749598408531, 5.17668933935653438606742872868, 6.59041494482097128670101060587, 7.61163366149491691430142579624, 8.623498336181693593648504763670, 9.264307654880326413208734649050, 10.38175946023813200149032295281, 11.09014772256919555372432895019

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