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Properties

Label 2-462-33.17-c1-0-3
Degree $2$
Conductor $462$
Sign $0.860 - 0.510i$
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 + (−1.26 − 1.18i)3-s + (−0.809 + 0.587i)4-s + (−4.04 − 1.31i)5-s + (−0.737 + 1.56i)6-s + (−0.587 − 0.809i)7-s + (0.809 + 0.587i)8-s + (0.189 + 2.99i)9-s + 4.25i·10-s + (−0.991 + 3.16i)11-s + (1.71 + 0.216i)12-s + (1.38 − 0.449i)13-s + (−0.587 + 0.809i)14-s + (3.55 + 6.45i)15-s + (0.309 − 0.951i)16-s + (1.02 − 3.15i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.729 − 0.684i)3-s + (−0.404 + 0.293i)4-s + (−1.80 − 0.587i)5-s + (−0.300 + 0.639i)6-s + (−0.222 − 0.305i)7-s + (0.286 + 0.207i)8-s + (0.0632 + 0.998i)9-s + 1.34i·10-s + (−0.298 + 0.954i)11-s + (0.496 + 0.0625i)12-s + (0.383 − 0.124i)13-s + (−0.157 + 0.216i)14-s + (0.916 + 1.66i)15-s + (0.0772 − 0.237i)16-s + (0.248 − 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.860 - 0.510i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.860 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236698 + 0.0649186i\)
\(L(\frac12)\) \(\approx\) \(0.236698 + 0.0649186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 1.18i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (0.991 - 3.16i)T \)
good5 \( 1 + (4.04 + 1.31i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.38 + 0.449i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.02 + 3.15i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.26 + 1.73i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.63iT - 23T^{2} \)
29 \( 1 + (7.71 - 5.60i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.07 - 6.39i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.11 - 1.53i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.55 - 2.58i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.85iT - 43T^{2} \)
47 \( 1 + (3.47 - 4.78i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.422 - 0.137i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.66 - 6.41i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (14.2 + 4.63i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.736T + 67T^{2} \)
71 \( 1 + (-8.22 - 2.67i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.33 + 1.84i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.55 + 0.829i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.71 - 8.35i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.590iT - 89T^{2} \)
97 \( 1 + (2.90 + 8.93i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.20795832093952507178358311412, −10.65108947054797920322279035979, −9.340645471278055609680263350779, −8.243235788351256534626692730792, −7.55097383179373117213680485632, −6.83363521046655281256649702754, −5.05296276258823460387654150977, −4.40730580225262817998594699105, −3.06466591904051645621409656434, −1.13209800350416693918674721924, 0.21906176390177295443388984857, 3.52092697216268329811457239230, 3.99385120975745855642402321959, 5.45869261912864835210390639716, 6.23274325841573344790443519946, 7.38806845477149855511447330824, 8.106279543573561285141149458952, 9.061248731039873419617042494838, 10.18578296618894546745565044499, 11.10334750620593396763040409677

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