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Properties

Label 2-462-21.5-c1-0-12
Degree $2$
Conductor $462$
Sign $0.259 - 0.965i$
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 + (−1.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.358 − 0.621i)5-s − 1.73·6-s + (2.62 + 0.358i)7-s + 0.999i·8-s + (1.5 − 2.59i)9-s + (0.621 − 0.358i)10-s + (0.866 − 0.5i)11-s + (−1.49 − 0.866i)12-s + 3.46i·13-s + (2.09 + 1.62i)14-s + 1.24i·15-s + (−0.5 + 0.866i)16-s + (3.31 + 5.74i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.866 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.160 − 0.277i)5-s − 0.707·6-s + (0.990 + 0.135i)7-s + 0.353i·8-s + (0.5 − 0.866i)9-s + (0.196 − 0.113i)10-s + (0.261 − 0.150i)11-s + (−0.433 − 0.250i)12-s + 0.960i·13-s + (0.558 + 0.433i)14-s + 0.320i·15-s + (−0.125 + 0.216i)16-s + (0.804 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.259 - 0.965i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36495 + 1.04633i\)
\(L(\frac12)\) \(\approx\) \(1.36495 + 1.04633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (-2.62 - 0.358i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.358 + 0.621i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-3.31 - 5.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.24 - 0.717i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.27 + 3.62i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + (7.24 - 4.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + (-0.358 + 0.621i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.34 + 4.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.63 + 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.62 + 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.74 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-1.24 + 0.717i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.62 - 2.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + (-5.91 + 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−11.32249234667047814998879866517, −10.59110619455533845365411109441, −9.484721749616069147284555944170, −8.468494395153796615033461313111, −7.44584469164427706202128743789, −6.20157627643537330334958817802, −5.60994805238274922101322570990, −4.53434635949363754453424964468, −3.81913319414480920537091197833, −1.70542825862264546918502707485, 1.13723168083381171976866477952, 2.56558367020640900664978578440, 4.16181280543537568085828383389, 5.32620390951704529542262109142, 5.79214523805086805135003305159, 7.25413066908227540970794525935, 7.66919532098970018995340080410, 9.259152664901869446394385234999, 10.47494152634127498748205066590, 10.89913000676365879645721970520

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