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Properties

Label 2-456-19.11-c1-0-4
Degree $2$
Conductor $456$
Sign $0.837 + 0.546i$
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 − 0.866i)3-s + (−0.347 + 0.601i)5-s + 0.305·7-s + (−0.499 − 0.866i)9-s + 4.82·11-s + (−0.5 − 0.866i)13-s + (0.347 + 0.601i)15-s + (3.75 − 6.51i)17-s + (3.06 + 3.10i)19-s + (0.152 − 0.264i)21-s + (−0.347 − 0.601i)23-s + (2.25 + 3.91i)25-s − 0.999·27-s + (−5.06 − 8.77i)29-s − 1.82·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.155 + 0.269i)5-s + 0.115·7-s + (−0.166 − 0.288i)9-s + 1.45·11-s + (−0.138 − 0.240i)13-s + (0.0896 + 0.155i)15-s + (0.911 − 1.57i)17-s + (0.702 + 0.711i)19-s + (0.0333 − 0.0577i)21-s + (−0.0724 − 0.125i)23-s + (0.451 + 0.782i)25-s − 0.192·27-s + (−0.940 − 1.62i)29-s − 0.327·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55936 - 0.463957i\)
\(L(\frac12)\) \(\approx\) \(1.55936 - 0.463957i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-3.06 - 3.10i)T \)
good5 \( 1 + (0.347 - 0.601i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.305T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.75 + 6.51i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.347 + 0.601i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.06 + 8.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 - 6.51T + 37T^{2} \)
41 \( 1 + (2.69 - 4.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.84 - 3.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.71 + 4.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.04 + 3.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.194 + 0.337i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.91 - 6.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.45 - 9.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.19 + 3.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.21 - 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.739T + 83T^{2} \)
89 \( 1 + (-0.411 - 0.712i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.45 - 9.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42845013184103647563663607789, −9.750563450523572967771480528543, −9.405239821469270859270874999922, −8.068142618009413927336370507497, −7.38584344969382784135168965855, −6.45797921799566457354811481636, −5.36732885438047691826820640368, −3.94391880622174462609989476933, −2.87354649692896317311034782538, −1.23892178352253258141244413606, 1.54824863182356726832785591831, 3.36406422696934501525334580603, 4.20356081832464200812901194295, 5.35743662369542022094202757015, 6.49267251043121847344510935112, 7.57212346497224088523306427759, 8.674145511591480982088907853004, 9.227848019525973638352670293941, 10.21774671879528914511141862732, 11.11286076003481146780075981643

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