Properties

Label 2-456-456.395-c0-0-1
Degree $2$
Conductor $456$
Sign $0.992 + 0.120i$
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  + (0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−1.70 + 0.984i)11-s + (0.766 − 0.642i)12-s + (0.173 + 0.984i)16-s + (0.592 − 1.62i)17-s + (−0.766 − 0.642i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−1.70 + 0.984i)11-s + (0.766 − 0.642i)12-s + (0.173 + 0.984i)16-s + (0.592 − 1.62i)17-s + (−0.766 − 0.642i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(0.227573\)
Root analytic conductor: \(0.477046\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :0),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400501036\)
\(L(\frac12)\) \(\approx\) \(1.400501036\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
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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

−11.58644033804605129304195688490, −10.67531827338755564451744493589, −9.371442260815242085057194653343, −8.042032750089009665627504528303, −7.47291632305022087515202235906, −6.75095672375867351603708165988, −5.50734219663028214667740218416, −4.79357235310370281409538590780, −3.07323738108844031390228529974, −2.22450802316928221900770167198, 2.40413955278088361217354288840, 3.48047389258595842018182408685, 4.40013635516332379903188780361, 5.58188382645740824915253304497, 6.05507416116499818873264579333, 7.81801130376256686063151760639, 8.541839343564402169167972820196, 9.995462462372261982147277781364, 10.57327597221300734412038679539, 11.02663714980886234298396075767

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