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Properties

Label 2-456-456.155-c0-0-1
Degree $2$
Conductor $456$
Sign $0.939 - 0.342i$
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  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)6-s + (−0.500 + 0.866i)8-s + (−0.499 − 0.866i)9-s + (−0.592 − 0.342i)11-s + (0.939 + 0.342i)12-s + (−0.939 + 0.342i)16-s + (−1.11 + 1.32i)17-s + (0.173 − 0.984i)18-s + (−0.173 − 0.984i)19-s + (−0.233 − 0.642i)22-s + (0.499 + 0.866i)24-s + (0.939 + 0.342i)25-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)6-s + (−0.500 + 0.866i)8-s + (−0.499 − 0.866i)9-s + (−0.592 − 0.342i)11-s + (0.939 + 0.342i)12-s + (−0.939 + 0.342i)16-s + (−1.11 + 1.32i)17-s + (0.173 − 0.984i)18-s + (−0.173 − 0.984i)19-s + (−0.233 − 0.642i)22-s + (0.499 + 0.866i)24-s + (0.939 + 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.227573\)
Root analytic conductor: \(0.477046\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.353084418\)
\(L(\frac12)\) \(\approx\) \(1.353084418\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
good5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
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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

−11.52368131895017795232506185629, −10.73628345340765688341690791189, −9.022462245455943840458792314166, −8.475363609847913099215385258326, −7.52655089942636547068638560053, −6.68620586852354494227201916161, −5.90595622501318476806678510516, −4.64134433053528326880573489310, −3.36803279194637224693476486498, −2.25422590194552494922479390195, 2.26060298859621526070966466616, 3.23744707804375481069570165369, 4.47522308060303567248952059583, 5.06870835415372396495147102034, 6.32169895766462468038192638720, 7.61040188355863166937882967244, 8.869404534321264847871137462647, 9.640535587750028630255993995325, 10.46835434950025684811188695550, 11.11566401672859286280262343885

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