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Properties

Label 2-456-456.371-c0-0-1
Degree $2$
Conductor $456$
Sign $0.877 + 0.479i$
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 + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (1.11 − 0.642i)11-s + (−0.939 − 0.342i)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 + (−0.173 + 0.984i)24-s + (−0.766 − 0.642i)25-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (1.11 − 0.642i)11-s + (−0.939 − 0.342i)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 + (−0.173 + 0.984i)24-s + (−0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(0.227573\)
Root analytic conductor: \(0.477046\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :0),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9314978412\)
\(L(\frac12)\) \(\approx\) \(0.9314978412\)
\(L(1)\) 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 \( 1 + (-0.766 - 0.642i)T \)
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

−11.27688028922704793334091761906, −10.23058347715233385500414472807, −9.475539906193342698303658864429, −8.750624527835112901274139890973, −8.089665006437769031637283727172, −6.63597500865994226887285714836, −5.04932523465199771566247850214, −4.07911500863910084336060449241, −3.21818330032895592244622420676, −1.90859637899668004557641393485, 1.70915252033712958453505294437, 3.57717016684326603828098256185, 4.65477982764100097920471162564, 6.08843723779040191813334854047, 6.93599250019069259339813607499, 7.52610850197381885973726946623, 8.670968385659675939929295035158, 9.213843311972728815636261830780, 10.01714259816695836277694633853, 11.52550825316153266248530674854

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