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Properties

Label 2-2268-63.58-c1-0-30
Degree $2$
Conductor $2268$
Sign $-0.888 - 0.458i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)5-s + (−0.5 + 2.59i)7-s + (1.5 − 2.59i)11-s + (−1 + 1.73i)13-s + (−1.5 − 2.59i)17-s + (0.5 − 0.866i)19-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + (3 + 5.19i)29-s − 7·31-s + (7.5 − 2.59i)35-s + (0.5 − 0.866i)37-s + (−3 + 5.19i)41-s + (2 + 3.46i)43-s − 9·47-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s + (−0.188 + 0.981i)7-s + (0.452 − 0.783i)11-s + (−0.277 + 0.480i)13-s + (−0.363 − 0.630i)17-s + (0.114 − 0.198i)19-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + (0.557 + 0.964i)29-s − 1.25·31-s + (1.26 − 0.439i)35-s + (0.0821 − 0.142i)37-s + (−0.468 + 0.811i)41-s + (0.304 + 0.528i)43-s − 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.682435589394762343993097391725, −8.067070738954450795111832135701, −7.00408895697570206789308721471, −6.20510743827946891125923125282, −5.24127588809493621386342214859, −4.68029688432423288141268881833, −3.68910135041117545302581857980, −2.67083263474190673979202311800, −1.37022987096987733962398149336, 0, 1.71461973223273899886713053790, 2.97132786522700562284428208741, 3.80662682966081054569339583484, 4.36567633114570737319857894131, 5.63746391515603009460602137897, 6.62655164966622257901573642173, 7.18031004327649279169844722785, 7.67982755718067335406529723644, 8.559667703884861700765028652580

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