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Properties

Label 2-456-57.50-c1-0-8
Degree $2$
Conductor $456$
Sign $0.823 - 0.567i$
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  + (1.68 + 0.396i)3-s + (−2.18 + 1.26i)5-s + 3.37·7-s + (2.68 + 1.33i)9-s − 3.46i·11-s + (2.87 + 1.65i)13-s + (−4.18 + 1.26i)15-s + (−2.18 + 1.26i)17-s + (−4 + 1.73i)19-s + (5.68 + 1.33i)21-s + (7.93 + 4.57i)23-s + (0.686 − 1.18i)25-s + (4 + 3.31i)27-s + (0.186 − 0.322i)29-s + 7.72i·31-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s + (−0.977 + 0.564i)5-s + 1.27·7-s + (0.895 + 0.445i)9-s − 1.04i·11-s + (0.796 + 0.459i)13-s + (−1.08 + 0.325i)15-s + (−0.530 + 0.306i)17-s + (−0.917 + 0.397i)19-s + (1.24 + 0.291i)21-s + (1.65 + 0.954i)23-s + (0.137 − 0.237i)25-s + (0.769 + 0.638i)27-s + (0.0345 − 0.0598i)29-s + 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81707 + 0.565660i\)
\(L(\frac12)\) \(\approx\) \(1.81707 + 0.565660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.68 - 0.396i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (2.18 - 1.26i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.37T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (-2.87 - 1.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.18 - 1.26i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.93 - 4.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.186 + 0.322i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.72iT - 31T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + (3.18 + 5.51i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.87 + 10.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.813 + 0.469i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.81 + 4.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.813 + 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.24 + 0.718i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.18 + 10.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.87 + 4.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.9 - 7.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (-0.813 + 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.44 + 1.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.95856862087742237413623216263, −10.63519306999175766706234540816, −8.832346017310680705663488584277, −8.639341030947806993576836540492, −7.64475544101979242707318398130, −6.83862966856864334618717282795, −5.25028038751468802472410700363, −4.02442529010585267171412503994, −3.34369411082870113551504688849, −1.75603747263669751300752846849, 1.34631190596867538276927062502, 2.79473975621977905721167040647, 4.39522139934633295859352812344, 4.66840920298270849449294116441, 6.61809709182501694191577846238, 7.60214248610931069808981348516, 8.344939199935375981913995514028, 8.770263100739206334220854111070, 10.00922049767684203165341428409, 11.15181284183085648124497490056

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