↑

Properties

Label 2-462-77.62-c1-0-12
Degree $2$
Conductor $462$
Sign $0.0113 + 0.999i$
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

Related objects

Downloads

Learn more

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 + (0.261 − 2.63i)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 + (−1.97 − 1.75i)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 + (0.0986 − 0.995i)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 + (−0.528 − 0.470i)14-s + (−0.0721 − 0.222i)15-s + (−0.202 + 0.146i)16-s + (−0.584 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0113 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0113 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0113 + 0.999i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0113 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39535 - 1.37967i\)
\(L(\frac12)\) \(\approx\) \(1.39535 - 1.37967i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.261 + 2.63i)T \)
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 \( 1 + (-2.47 + 3.39i)T + (-29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.83470287102886262813070183405, −10.06641768507017917400753261317, −9.103823413774945666309499315044, −8.264363309495731127607256008684, −7.17056270690587100728291133924, −6.07325179127044203932287537890, −4.56471656415001030443619329740, −4.02800995054180970550157075489, −2.79592880143629030651549970232, −1.12224014768791912405074539510, 2.14370487784455506406169650682, 3.36771143786131584871061502308, 4.54668174673710007173314677396, 5.64963822902879966393738636807, 6.85756338036016649762071967390, 7.33955689000129921928396260973, 8.782845249350929580135469319263, 8.982938959061058894943679774884, 10.28477274600029733261383850249, 11.67230754931192915220922311335

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