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Properties

Label 2-462-33.32-c1-0-5
Degree $2$
Conductor $462$
Sign $0.535 - 0.844i$
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  + 2-s + (−1.64 + 0.543i)3-s + 4-s + 0.118i·5-s + (−1.64 + 0.543i)6-s + i·7-s + 8-s + (2.40 − 1.78i)9-s + 0.118i·10-s + (2.10 + 2.56i)11-s + (−1.64 + 0.543i)12-s + 0.913i·13-s + i·14-s + (−0.0644 − 0.195i)15-s + 16-s − 0.404·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.949 + 0.313i)3-s + 0.5·4-s + 0.0530i·5-s + (−0.671 + 0.221i)6-s + 0.377i·7-s + 0.353·8-s + (0.803 − 0.595i)9-s + 0.0375i·10-s + (0.633 + 0.773i)11-s + (−0.474 + 0.156i)12-s + 0.253i·13-s + 0.267i·14-s + (−0.0166 − 0.0503i)15-s + 0.250·16-s − 0.0980·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42395 + 0.782759i\)
\(L(\frac12)\) \(\approx\) \(1.42395 + 0.782759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.64 - 0.543i)T \)
7 \( 1 - iT \)
11 \( 1 + (-2.10 - 2.56i)T \)
good5 \( 1 - 0.118iT - 5T^{2} \)
13 \( 1 - 0.913iT - 13T^{2} \)
17 \( 1 + 0.404T + 17T^{2} \)
19 \( 1 - 7.70iT - 19T^{2} \)
23 \( 1 - 1.37iT - 23T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 8.74T + 31T^{2} \)
37 \( 1 - 4.13T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 2.85iT - 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 8.81iT - 53T^{2} \)
59 \( 1 + 5.49iT - 59T^{2} \)
61 \( 1 - 7.30iT - 61T^{2} \)
67 \( 1 + 2.01T + 67T^{2} \)
71 \( 1 - 14.2iT - 71T^{2} \)
73 \( 1 + 11.7iT - 73T^{2} \)
79 \( 1 + 4.34iT - 79T^{2} \)
83 \( 1 - 0.700T + 83T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 + 5.48T + 97T^{2} \)
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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

−11.51557833208151669249711643724, −10.31399092824636594820769769859, −9.788131629167261575846023217975, −8.467173352333440962619836919482, −7.11940538703613205342526333786, −6.38218263042461006748189517429, −5.45644544183366201766916517788, −4.52753583720986494986404455707, −3.54670214590944533217088036223, −1.71498882790439862937216128484, 1.00757991842127431477116312855, 2.86910226449255136742788036112, 4.34355447177741993846801081054, 5.11244278257712781764385153660, 6.30326632548026472280784696795, 6.80904236732147083468494200797, 7.931609199615209279454617129744, 9.167334585557930447981482148884, 10.41476079322099058752172560959, 11.13565367801848478070012005798

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