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Properties

Label 2-456-57.50-c1-0-3
Degree $2$
Conductor $456$
Sign $-0.903 - 0.429i$
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  + (0.5 + 1.65i)3-s + (−0.686 + 0.396i)5-s − 2.37·7-s + (−2.5 + 1.65i)9-s + 3.46i·11-s + (−2.87 − 1.65i)13-s + (−1 − 0.939i)15-s + (−0.686 + 0.396i)17-s + (−4 + 1.73i)19-s + (−1.18 − 3.93i)21-s + (6.43 + 3.71i)23-s + (−2.18 + 3.78i)25-s + (−4 − 3.31i)27-s + (2.68 − 4.65i)29-s + 4.40i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.957i)3-s + (−0.306 + 0.177i)5-s − 0.896·7-s + (−0.833 + 0.552i)9-s + 1.04i·11-s + (−0.796 − 0.459i)13-s + (−0.258 − 0.242i)15-s + (−0.166 + 0.0960i)17-s + (−0.917 + 0.397i)19-s + (−0.258 − 0.858i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.903 - 0.429i$
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: \((2,\ 456,\ (\ :1/2),\ -0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184500 + 0.818129i\)
\(L(\frac12)\) \(\approx\) \(0.184500 + 0.818129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 1.65i)T \)
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

−11.25160589138825374236945659145, −10.39976172696792144956505773157, −9.689783265781493089883582045508, −9.029780217906311135531475898678, −7.81635743040425189823939546537, −6.94212320066562890547445484708, −5.65189682576313872526377181740, −4.59192207476172909818066895755, −3.59138056333536249795131425699, −2.49580105845803161955352738847, 0.47655997436487471427521096151, 2.41786027175302924978496891864, 3.44945061038608635755683600402, 4.93741285705684718241982912825, 6.40996959535887986007026788228, 6.75605705823130882759737364384, 8.035240157314240908290777466980, 8.733809972374540646217511380965, 9.600616623716319254878919408470, 10.84634307165241503901118149542

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