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Properties

Label 2-456-152.75-c1-0-33
Degree $2$
Conductor $456$
Sign $-0.967 - 0.253i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
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  + (−0.612 − 1.27i)2-s − i·3-s + (−1.25 + 1.56i)4-s − 4.23i·5-s + (−1.27 + 0.612i)6-s − 3.84i·7-s + (2.75 + 0.637i)8-s − 9-s + (−5.39 + 2.59i)10-s + 2.95·11-s + (1.56 + 1.25i)12-s + 1.09·13-s + (−4.90 + 2.35i)14-s − 4.23·15-s + (−0.873 − 3.90i)16-s + 2.81·17-s + ⋯
L(s)  = 1  + (−0.432 − 0.901i)2-s − 0.577i·3-s + (−0.625 + 0.780i)4-s − 1.89i·5-s + (−0.520 + 0.249i)6-s − 1.45i·7-s + (0.974 + 0.225i)8-s − 0.333·9-s + (−1.70 + 0.819i)10-s + 0.889·11-s + (0.450 + 0.360i)12-s + 0.304·13-s + (−1.31 + 0.629i)14-s − 1.09·15-s + (−0.218 − 0.975i)16-s + 0.682·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.967 - 0.253i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ -0.967 - 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135361 + 1.05160i\)
\(L(\frac12)\) \(\approx\) \(0.135361 + 1.05160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.612 + 1.27i)T \)
3 \( 1 + iT \)
19 \( 1 + (0.124 - 4.35i)T \)
good5 \( 1 + 4.23iT - 5T^{2} \)
7 \( 1 + 3.84iT - 7T^{2} \)
11 \( 1 - 2.95T + 11T^{2} \)
13 \( 1 - 1.09T + 13T^{2} \)
17 \( 1 - 2.81T + 17T^{2} \)
23 \( 1 - 2.47iT - 23T^{2} \)
29 \( 1 - 7.63T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + 0.594T + 37T^{2} \)
41 \( 1 + 5.98iT - 41T^{2} \)
43 \( 1 - 8.62T + 43T^{2} \)
47 \( 1 - 8.38iT - 47T^{2} \)
53 \( 1 - 1.76T + 53T^{2} \)
59 \( 1 - 7.19iT - 59T^{2} \)
61 \( 1 + 1.92iT - 61T^{2} \)
67 \( 1 - 11.9iT - 67T^{2} \)
71 \( 1 + 2.10T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 4.29T + 79T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 + 14.7iT - 89T^{2} \)
97 \( 1 + 4.36iT - 97T^{2} \)
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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

−10.56767962137289389141914052839, −9.627913372540740065657729443963, −8.862421422394932885573935634119, −8.036196650264856977149936886132, −7.29079571013358467184661212889, −5.66992290857884070590468625462, −4.38871721392165734424900115197, −3.72176855764429714507680754008, −1.51950690066002513218597842614, −0.872582420294433104722124514892, 2.44595324643079802273673372810, 3.67946264941704396634754436959, 5.19947890001265697376545759777, 6.23048272542628715545921936473, 6.71769206026615558823954437152, 7.893466636309913400876867372608, 8.954495537059875078486959180635, 9.594537808482349204686429963985, 10.56852021949839438388574355811, 11.22277559441531800273923568023

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