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Properties

Label 2-462-21.17-c1-0-11
Degree $2$
Conductor $462$
Sign $0.932 - 0.360i$
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  + (0.866 − 0.5i)2-s + (1.10 + 1.33i)3-s + (0.499 − 0.866i)4-s + (−0.241 − 0.417i)5-s + (1.62 + 0.599i)6-s + (1.48 + 2.19i)7-s − 0.999i·8-s + (−0.548 + 2.94i)9-s + (−0.417 − 0.241i)10-s + (0.866 + 0.5i)11-s + (1.70 − 0.292i)12-s + 0.550i·13-s + (2.38 + 1.15i)14-s + (0.289 − 0.783i)15-s + (−0.5 − 0.866i)16-s + (0.926 − 1.60i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.639 + 0.769i)3-s + (0.249 − 0.433i)4-s + (−0.107 − 0.186i)5-s + (0.663 + 0.244i)6-s + (0.560 + 0.827i)7-s − 0.353i·8-s + (−0.182 + 0.983i)9-s + (−0.132 − 0.0762i)10-s + (0.261 + 0.150i)11-s + (0.492 − 0.0845i)12-s + 0.152i·13-s + (0.636 + 0.308i)14-s + (0.0747 − 0.202i)15-s + (−0.125 − 0.216i)16-s + (0.224 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42181 + 0.451140i\)
\(L(\frac12)\) \(\approx\) \(2.42181 + 0.451140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-1.10 - 1.33i)T \)
7 \( 1 + (-1.48 - 2.19i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (0.241 + 0.417i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.550iT - 13T^{2} \)
17 \( 1 + (-0.926 + 1.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.334 - 0.192i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.216 - 0.124i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.48iT - 29T^{2} \)
31 \( 1 + (-1.08 - 0.623i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.72 + 4.72i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 + 0.682T + 43T^{2} \)
47 \( 1 + (-4.41 - 7.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.54 + 3.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.82 + 3.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.63 - 4.40i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.35 + 7.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.76iT - 71T^{2} \)
73 \( 1 + (1.94 + 1.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.71 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.48T + 83T^{2} \)
89 \( 1 + (-5.64 - 9.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.8iT - 97T^{2} \)
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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

−11.17026723372616321439768844215, −10.24371255563319802107307581834, −9.338636257471992281091623642317, −8.580872854556889139623697014972, −7.61212092448782684593006361336, −6.14575512930120830261569760621, −5.05421707610393016090292655945, −4.34836426094996912051003754666, −3.11231712120964679204889742729, −2.00596738668844378202834183046, 1.48887649104982473267792228035, 3.08159719405166803336931288174, 4.02094516700785826859437269233, 5.31892018841728404409104855532, 6.59468675590029821431525379675, 7.22418878886224225012814475376, 8.083768145755378900342768049393, 8.856138649275349912570284407892, 10.18775204301489402904762581379, 11.20113359908912392962138946346

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