Properties

Label 2-456-456.11-c1-0-28
Degree $2$
Conductor $456$
Sign $0.999 + 0.0279i$
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  + (−1.22 + 0.703i)2-s + (−1.34 − 1.08i)3-s + (1.01 − 1.72i)4-s + (1.58 − 2.74i)5-s + (2.41 + 0.388i)6-s + 3.56i·7-s + (−0.0261 + 2.82i)8-s + (0.629 + 2.93i)9-s + (−0.0138 + 4.47i)10-s + 2.09i·11-s + (−3.24 + 1.22i)12-s + (1.68 − 0.972i)13-s + (−2.50 − 4.37i)14-s + (−5.11 + 1.97i)15-s + (−1.95 − 3.48i)16-s + (3.17 + 1.83i)17-s + ⋯
L(s)  = 1  + (−0.867 + 0.497i)2-s + (−0.777 − 0.628i)3-s + (0.505 − 0.862i)4-s + (0.708 − 1.22i)5-s + (0.987 + 0.158i)6-s + 1.34i·7-s + (−0.00925 + 0.999i)8-s + (0.209 + 0.977i)9-s + (−0.00436 + 1.41i)10-s + 0.631i·11-s + (−0.935 + 0.353i)12-s + (0.467 − 0.269i)13-s + (−0.670 − 1.16i)14-s + (−1.32 + 0.508i)15-s + (−0.489 − 0.872i)16-s + (0.770 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0279i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.999 + 0.0279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863480 - 0.0120770i\)
\(L(\frac12)\) \(\approx\) \(0.863480 - 0.0120770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.703i)T \)
3 \( 1 + (1.34 + 1.08i)T \)
19 \( 1 + (-4.34 - 0.328i)T \)
good5 \( 1 + (-1.58 + 2.74i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.56iT - 7T^{2} \)
11 \( 1 - 2.09iT - 11T^{2} \)
13 \( 1 + (-1.68 + 0.972i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.17 - 1.83i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.948 - 1.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.681 + 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.65iT - 31T^{2} \)
37 \( 1 + 4.67iT - 37T^{2} \)
41 \( 1 + (-3.85 - 2.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.07 - 7.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.04 - 6.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.56 + 2.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.42 - 0.821i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.78 + 3.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.07 - 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.04 + 13.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.66 + 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.5 + 7.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.96iT - 83T^{2} \)
89 \( 1 + (9.31 - 5.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 - 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.10782176048269560593321151142, −9.790839958377129909777818432689, −9.343057879819924499149384506907, −8.284794225154026894020186292867, −7.58651337290719448148211717671, −6.15037483366042909395701034857, −5.66878998092215778214754707134, −4.96634463101109317139378215171, −2.20902616594171058429939530064, −1.14592819160512055969806366543, 1.04911895840229017057229075306, 3.06303831386395159543914066145, 3.85041168585389433775978676919, 5.49857408839433785533153966501, 6.81568888958819829534042361302, 7.07828701054249554048226151144, 8.573165543849916405403959797590, 9.842650084869045228962024418229, 10.14640733292226554258111488963, 10.95115051750739119898694159869

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