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Properties

Label 2-456-19.7-c1-0-8
Degree $2$
Conductor $456$
Sign $-0.949 + 0.312i$
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 + (−1.53 − 2.65i)5-s − 2.06·7-s + (−0.499 + 0.866i)9-s − 6.45·11-s + (−0.5 + 0.866i)13-s + (1.53 − 2.65i)15-s + (−0.694 − 1.20i)17-s + (−3.75 + 2.20i)19-s + (−1.03 − 1.78i)21-s + (−1.53 + 2.65i)23-s + (−2.19 + 3.80i)25-s − 0.999·27-s + (1.75 − 3.04i)29-s + 9.45·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.685 − 1.18i)5-s − 0.780·7-s + (−0.166 + 0.288i)9-s − 1.94·11-s + (−0.138 + 0.240i)13-s + (0.395 − 0.685i)15-s + (−0.168 − 0.291i)17-s + (−0.862 + 0.506i)19-s + (−0.225 − 0.390i)21-s + (−0.319 + 0.553i)23-s + (−0.438 + 0.760i)25-s − 0.192·27-s + (0.326 − 0.565i)29-s + 1.69·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.949 + 0.312i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ -0.949 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0328924 - 0.205053i\)
\(L(\frac12)\) \(\approx\) \(0.0328924 - 0.205053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.75 - 2.20i)T \)
good5 \( 1 + (1.53 + 2.65i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.06T + 7T^{2} \)
11 \( 1 + 6.45T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.694 + 1.20i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.53 - 2.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.45T + 31T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 + (5.06 + 8.77i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.03 + 5.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.29 + 9.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.59 - 9.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.56 - 4.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.72 - 2.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.36 + 5.83i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.56 - 7.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.790 - 1.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 + (5.22 - 9.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.36 + 5.83i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.35614767939240507248190967537, −9.939120036047705182557420020728, −8.613298294895063598749641612871, −8.275618158209599077507229718362, −7.14220279522677276995458851228, −5.65155214137333701645396386067, −4.80470271464506886205789532181, −3.82655522276379584557257224375, −2.49383525775058505295071150770, −0.11406657696082868091680608505, 2.61221931917186251718164038700, 3.13279296713759848666879749541, 4.67809435462431667509939511686, 6.18402846570298184814269656191, 6.88192084930186325242792634316, 7.83130960799737614559114144038, 8.442487519269947365078794858077, 9.967661753188984370339960019607, 10.53681277428000919303759571946, 11.34855740634881041727761037095

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