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Properties

Label 2-462-33.2-c1-0-18
Degree $2$
Conductor $462$
Sign $-0.908 + 0.418i$
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 + (−1.72 + 0.188i)3-s + (−0.809 − 0.587i)4-s + (0.152 − 0.0496i)5-s + (−0.352 + 1.69i)6-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s + (2.92 − 0.650i)9-s − 0.160i·10-s + (3.12 − 1.11i)11-s + (1.50 + 0.859i)12-s + (−6.36 − 2.06i)13-s + (−0.587 − 0.809i)14-s + (−0.253 + 0.114i)15-s + (0.309 + 0.951i)16-s + (−0.841 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.994 + 0.109i)3-s + (−0.404 − 0.293i)4-s + (0.0683 − 0.0221i)5-s + (−0.143 + 0.692i)6-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s + (0.976 − 0.216i)9-s − 0.0507i·10-s + (0.941 − 0.337i)11-s + (0.434 + 0.248i)12-s + (−1.76 − 0.573i)13-s + (−0.157 − 0.216i)14-s + (−0.0654 + 0.0295i)15-s + (0.0772 + 0.237i)16-s + (−0.204 − 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159763 - 0.728720i\)
\(L(\frac12)\) \(\approx\) \(0.159763 - 0.728720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.188i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-3.12 + 1.11i)T \)
good5 \( 1 + (-0.152 + 0.0496i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (6.36 + 2.06i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.841 + 2.59i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.45 + 4.76i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.94iT - 23T^{2} \)
29 \( 1 + (5.12 + 3.72i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.60 + 8.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.501 + 0.364i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.44 + 3.23i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.41iT - 43T^{2} \)
47 \( 1 + (-1.56 - 2.15i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.46 + 0.801i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.48 + 11.6i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.68 - 2.49i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.38T + 67T^{2} \)
71 \( 1 + (-10.0 + 3.25i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.0597 + 0.0822i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (8.72 + 2.83i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.39 - 13.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (3.20 - 9.85i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.02754235139102549789314855932, −9.677379589247348916609222382866, −9.502897225351791688873446853584, −7.78314285953776746842191707930, −6.85269998070229378546898492592, −5.70884016641423969837719987799, −4.81108576122957269427360213040, −3.91565948739142931837121079192, −2.24409453943415380052669054214, −0.47695820140610313237088975076, 1.93042244200188327945597409322, 4.07700521018058763763999854708, 4.84440693022425598411879899095, 5.91427979748191519830448504688, 6.70624783289717987495016100973, 7.47953532340802849880159270363, 8.649392090432689401712178211272, 9.719396593868481019902823919746, 10.47335414035735714846407545763, 11.72094469742388442412576480531

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