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Properties

Label 2-462-77.41-c1-0-9
Degree $2$
Conductor $462$
Sign $-0.753 + 0.657i$
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  + (−0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−2.23 + 3.08i)5-s + (0.809 + 0.587i)6-s + (−1.26 + 2.32i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + 3.80·10-s + (−2.45 − 2.23i)11-s − i·12-s + (3.55 − 2.58i)13-s + (2.62 − 0.345i)14-s + (1.17 − 3.62i)15-s + (−0.809 − 0.587i)16-s + (−3.02 − 2.20i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (−1.00 + 1.37i)5-s + (0.330 + 0.239i)6-s + (−0.477 + 0.878i)7-s + (0.336 − 0.109i)8-s + (0.269 − 0.195i)9-s + 1.20·10-s + (−0.738 − 0.673i)11-s − 0.288i·12-s + (0.987 − 0.717i)13-s + (0.701 − 0.0924i)14-s + (0.303 − 0.935i)15-s + (−0.202 − 0.146i)16-s + (−0.734 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0467982 - 0.124747i\)
\(L(\frac12)\) \(\approx\) \(0.0467982 - 0.124747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 - 2.32i)T \)
11 \( 1 + (2.45 + 2.23i)T \)
good5 \( 1 + (2.23 - 3.08i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-3.55 + 2.58i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.02 + 2.20i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.91 + 5.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.81T + 23T^{2} \)
29 \( 1 + (-5.96 - 1.93i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.39 - 6.04i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.733 - 2.25i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.59 + 4.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.46iT - 43T^{2} \)
47 \( 1 + (9.32 - 3.03i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.843 + 0.612i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (8.59 + 2.79i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.48 + 1.80i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.07T + 67T^{2} \)
71 \( 1 + (12.3 + 8.96i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.732 - 2.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.32 + 5.95i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.86 + 1.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.42iT - 89T^{2} \)
97 \( 1 + (6.55 + 9.01i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.80668617029541782635261047519, −10.21345672219128282326736833963, −8.854229435772374229172308679827, −8.159475584295645516338115012330, −6.93084008146217473458590272080, −6.22322060455595248132519445180, −4.80220441266701217422489295954, −3.31779833697154125058213715113, −2.76507636250323143568970310473, −0.10738137554753008637978005655, 1.38244053666108397743240568783, 4.08547026784577426388853836107, 4.55309091022887126090043084462, 5.92751272029328636771577645019, 6.81762634863198831328208996104, 7.989076367193534099932901275239, 8.317154920253508461847923521888, 9.576134816769880621108716275881, 10.41009706460988565695377077188, 11.38876899319243395400665824274

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