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Properties

Label 2-456-456.59-c0-0-1
Degree $2$
Conductor $456$
Sign $-0.660 + 0.750i$
Analytic cond. $0.227573$
Root an. cond. $0.477046$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (−0.5 + 0.866i)8-s + (−0.499 − 0.866i)9-s + (−1.11 − 0.642i)11-s + (−0.766 + 0.642i)12-s + (0.766 + 0.642i)16-s + (1.70 + 0.300i)17-s + (−0.939 + 0.342i)18-s + (0.939 + 0.342i)19-s + (−0.826 + 0.984i)22-s + (0.499 + 0.866i)24-s + (−0.766 + 0.642i)25-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (−0.5 + 0.866i)8-s + (−0.499 − 0.866i)9-s + (−1.11 − 0.642i)11-s + (−0.766 + 0.642i)12-s + (0.766 + 0.642i)16-s + (1.70 + 0.300i)17-s + (−0.939 + 0.342i)18-s + (0.939 + 0.342i)19-s + (−0.826 + 0.984i)22-s + (0.499 + 0.866i)24-s + (−0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(0.227573\)
Root analytic conductor: \(0.477046\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :0),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9445455834\)
\(L(\frac12)\) \(\approx\) \(0.9445455834\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
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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

−11.15864246976139054047640263842, −10.04178408420740084531060107803, −9.357507764602641671521489393965, −8.079504402031101372185175920781, −7.75263135321376833409628585809, −6.04088376688487035022803926177, −5.26240159342389655124947036482, −3.56260993936703421325136290140, −2.82214362158663499074001965395, −1.33980754589844892023046833427, 2.84641460929231240839944510222, 3.97871376781667324877004102999, 5.11173622023522950469947935138, 5.67384441544092704610410892857, 7.32574568702027908069721617115, 7.84793018454428025514232021504, 8.813893367100100303387830502184, 9.878099042661352745938216392241, 10.18221068852478701566736367240, 11.67515660697576508496265237737

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