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Properties

Label 2-468-117.94-c1-0-3
Degree $2$
Conductor $468$
Sign $-0.427 - 0.904i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (−1.43 + 0.970i)3-s + (−0.883 + 1.52i)5-s + 3.13·7-s + (1.11 − 2.78i)9-s + (−0.284 + 0.492i)11-s + (1.34 + 3.34i)13-s + (−0.216 − 3.05i)15-s + (−0.233 + 0.404i)17-s + (−3.16 + 5.48i)19-s + (−4.49 + 3.03i)21-s − 9.40·23-s + (0.939 + 1.62i)25-s + (1.09 + 5.07i)27-s + (−1.69 + 2.93i)29-s + (2.42 − 4.19i)31-s + ⋯
L(s)  = 1  + (−0.828 + 0.560i)3-s + (−0.395 + 0.684i)5-s + 1.18·7-s + (0.372 − 0.927i)9-s + (−0.0858 + 0.148i)11-s + (0.371 + 0.928i)13-s + (−0.0559 − 0.788i)15-s + (−0.0566 + 0.0980i)17-s + (−0.726 + 1.25i)19-s + (−0.980 + 0.662i)21-s − 1.96·23-s + (0.187 + 0.325i)25-s + (0.211 + 0.977i)27-s + (−0.314 + 0.544i)29-s + (0.434 − 0.753i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.427 - 0.904i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ -0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501173 + 0.791358i\)
\(L(\frac12)\) \(\approx\) \(0.501173 + 0.791358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.43 - 0.970i)T \)
13 \( 1 + (-1.34 - 3.34i)T \)
good5 \( 1 + (0.883 - 1.52i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.13T + 7T^{2} \)
11 \( 1 + (0.284 - 0.492i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.233 - 0.404i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.16 - 5.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 9.40T + 23T^{2} \)
29 \( 1 + (1.69 - 2.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.42 + 4.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.936 - 1.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.09T + 41T^{2} \)
43 \( 1 + 8.82T + 43T^{2} \)
47 \( 1 + (-5.16 - 8.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 + (5.02 + 8.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 - 4.58T + 67T^{2} \)
71 \( 1 + (5.25 - 9.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.76T + 73T^{2} \)
79 \( 1 + (4.23 + 7.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.06 + 1.83i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.53 + 4.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.79T + 97T^{2} \)
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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

−11.41015688113530362981248827678, −10.54186698153317297930820845389, −9.822504909118407046668047062550, −8.552790430329063075381717232388, −7.67180019519108244253292736896, −6.55449117343147242605926981554, −5.70130998778088377768979922635, −4.46681400250804645255193705003, −3.78379259426910884908344448203, −1.81315523039591897230588013608, 0.65770618125649766379763565464, 2.14931559020659546046714506988, 4.20779780460131317435094241970, 5.04892419265522210015661766026, 5.91847674891646164498504321880, 7.10456435677181165769838211716, 8.171872399501105032969378678694, 8.492569255324894682552657831288, 10.13551867868261879907830294870, 10.91751790339612668157677567226

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