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Properties

Label 2-456-57.8-c1-0-0
Degree $2$
Conductor $456$
Sign $-0.999 - 0.0179i$
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  + (−1.18 + 1.26i)3-s + (0.686 + 0.396i)5-s − 2.37·7-s + (−0.186 − 2.99i)9-s + 3.46i·11-s + (−2.87 + 1.65i)13-s + (−1.31 + 0.396i)15-s + (0.686 + 0.396i)17-s + (−4 − 1.73i)19-s + (2.81 − 2.99i)21-s + (−6.43 + 3.71i)23-s + (−2.18 − 3.78i)25-s + (4.00 + 3.31i)27-s + (−2.68 − 4.65i)29-s − 4.40i·31-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)3-s + (0.306 + 0.177i)5-s − 0.896·7-s + (−0.0620 − 0.998i)9-s + 1.04i·11-s + (−0.796 + 0.459i)13-s + (−0.339 + 0.102i)15-s + (0.166 + 0.0960i)17-s + (−0.917 − 0.397i)19-s + (0.614 − 0.653i)21-s + (−1.34 + 0.774i)23-s + (−0.437 − 0.757i)25-s + (0.769 + 0.638i)27-s + (−0.498 − 0.863i)29-s − 0.790i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.999 - 0.0179i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ -0.999 - 0.0179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00337970 + 0.377551i\)
\(L(\frac12)\) \(\approx\) \(0.00337970 + 0.377551i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (2.87 - 1.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.686 - 0.396i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.43 - 3.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.68 + 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.40iT - 31T^{2} \)
37 \( 1 - 7.86iT - 37T^{2} \)
41 \( 1 + (0.313 - 0.543i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.127 - 0.221i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.68 - 2.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.68 - 9.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.2 + 5.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.31 - 5.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.87 + 4.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.98 - 5.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-3.68 - 6.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.0 - 6.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.59380367179196043461764746233, −10.33288410158501084856962757918, −9.845020747186979387989689205349, −9.277840743998620954644863584834, −7.77676167283076151248547662878, −6.60651387671210076274568049346, −6.00181919454474874307956555416, −4.72533244877282792710544312305, −3.87750940412423696688896881007, −2.31551640330033601826994633281, 0.23585980539511422680049855115, 2.09225977665157019408925212517, 3.54493410165920295391710061150, 5.16493751875721731644196150026, 5.97161491155794860244535191658, 6.73280321544057648036614571663, 7.79961272524286964433345407049, 8.732047198318836855777476358238, 9.916191581840762997430188249017, 10.62983457972770903570389625140

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