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Properties

Label 2-462-77.41-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.955 - 0.294i$
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 + (−2.18 + 3.01i)5-s + (0.809 + 0.587i)6-s + (−2.63 − 0.190i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s − 3.72·10-s + (−2.71 − 1.90i)11-s + i·12-s + (−4.10 + 2.98i)13-s + (−1.39 − 2.24i)14-s + (−1.14 + 3.53i)15-s + (−0.809 − 0.587i)16-s + (4.90 + 3.56i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 + 0.475i)4-s + (−0.978 + 1.34i)5-s + (0.330 + 0.239i)6-s + (−0.997 − 0.0719i)7-s + (−0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s − 1.17·10-s + (−0.819 − 0.573i)11-s + 0.288i·12-s + (−1.13 + 0.828i)13-s + (−0.373 − 0.600i)14-s + (−0.296 + 0.913i)15-s + (−0.202 − 0.146i)16-s + (1.18 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.955 - 0.294i$
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: \((2,\ 462,\ (\ :1/2),\ -0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158572 + 1.05139i\)
\(L(\frac12)\) \(\approx\) \(0.158572 + 1.05139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.190i)T \)
11 \( 1 + (2.71 + 1.90i)T \)
good5 \( 1 + (2.18 - 3.01i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (4.10 - 2.98i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.90 - 3.56i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0255 + 0.0785i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.24T + 23T^{2} \)
29 \( 1 + (-9.40 - 3.05i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.60 + 2.20i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.49 - 7.68i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.40 - 7.41i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.56iT - 43T^{2} \)
47 \( 1 + (9.34 - 3.03i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.23 + 0.894i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.05 + 0.993i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.40 - 2.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (-5.72 - 4.16i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.831 + 2.55i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.41 + 8.82i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.69 - 7.04i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (4.43 + 6.09i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.61153602465788328840361442836, −10.45764293025147020837069496595, −9.758985472040883522310210044714, −8.380504890070822034627159086240, −7.65286441455025644889551231600, −6.89378848256584383428388989517, −6.18753553521202878828310317929, −4.60837696805683644194672467469, −3.29880969998032860139226228453, −2.90274207745045178815831995043, 0.52182976764306839265849964975, 2.62921351707391046266999828314, 3.60831059987304238162424085011, 4.80423712284399491521728552404, 5.37026451552482555630972763183, 7.20897804166225535026207875926, 7.967894643354987908802818370142, 9.000306186364894513376078722555, 9.801407203909267997781842505277, 10.46353191388238708665354070478

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