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Properties

Label 2-462-33.8-c1-0-11
Degree $2$
Conductor $462$
Sign $0.886 - 0.462i$
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  + (0.809 − 0.587i)2-s + (1.60 + 0.655i)3-s + (0.309 − 0.951i)4-s + (−2.04 + 2.81i)5-s + (1.68 − 0.411i)6-s + (0.951 + 0.309i)7-s + (−0.309 − 0.951i)8-s + (2.13 + 2.10i)9-s + 3.47i·10-s + (2.97 + 1.46i)11-s + (1.11 − 1.32i)12-s + (−2.76 − 3.80i)13-s + (0.951 − 0.309i)14-s + (−5.12 + 3.16i)15-s + (−0.809 − 0.587i)16-s + (5.89 + 4.28i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.925 + 0.378i)3-s + (0.154 − 0.475i)4-s + (−0.913 + 1.25i)5-s + (0.686 − 0.168i)6-s + (0.359 + 0.116i)7-s + (−0.109 − 0.336i)8-s + (0.713 + 0.700i)9-s + 1.09i·10-s + (0.897 + 0.441i)11-s + (0.323 − 0.381i)12-s + (−0.767 − 1.05i)13-s + (0.254 − 0.0825i)14-s + (−1.32 + 0.818i)15-s + (−0.202 − 0.146i)16-s + (1.43 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28656 + 0.560607i\)
\(L(\frac12)\) \(\approx\) \(2.28656 + 0.560607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-1.60 - 0.655i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-2.97 - 1.46i)T \)
good5 \( 1 + (2.04 - 2.81i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.76 + 3.80i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.89 - 4.28i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.19 - 0.711i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.430iT - 23T^{2} \)
29 \( 1 + (-0.591 + 1.82i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.01 - 0.740i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.840 + 2.58i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.21 + 6.82i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + (10.3 - 3.36i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.44 + 6.11i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (7.64 + 2.48i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.23 + 7.20i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.79T + 67T^{2} \)
71 \( 1 + (-8.84 + 12.1i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.238 - 0.0775i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.38 - 6.03i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.9 - 9.42i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 + (-3.63 + 2.64i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.95156668422381490943158418957, −10.39100827573267888795977207588, −9.599871788593350070946788085131, −8.197709921404162010348158398419, −7.59665451881472263389907303867, −6.57225970923605939271697238533, −5.12916526760166197001199981866, −3.83341680431352942107862292052, −3.36106084351594055773043567854, −2.08973388724685918313020050183, 1.33310948910784424221879505496, 3.16044578460613452379460444846, 4.26287491117472823267260623054, 4.91796458473284037343473598964, 6.48266131002577640594321941673, 7.50180203286266224090015463785, 8.108806153546674984566398376778, 8.978688593812530434677067342634, 9.659511031749948050652960924952, 11.59629997103982720428192131675

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