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Properties

Label 2-2240-280.53-c0-0-1
Degree $2$
Conductor $2240$
Sign $-0.167 - 0.985i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)3-s + (−0.258 + 0.965i)5-s + (0.707 − 0.707i)7-s + (−0.866 + 0.5i)9-s + (0.866 + 0.5i)11-s + (0.707 − 0.707i)13-s − 1.41·15-s + (0.5 + 0.866i)19-s + (1.22 + 0.707i)21-s + (−0.258 + 0.965i)23-s + (−0.866 − 0.499i)25-s − 1.41·29-s + (0.707 − 1.22i)31-s + (−0.366 + 1.36i)33-s + (0.500 + 0.866i)35-s + ⋯
L(s)  = 1  + (0.366 + 1.36i)3-s + (−0.258 + 0.965i)5-s + (0.707 − 0.707i)7-s + (−0.866 + 0.5i)9-s + (0.866 + 0.5i)11-s + (0.707 − 0.707i)13-s − 1.41·15-s + (0.5 + 0.866i)19-s + (1.22 + 0.707i)21-s + (−0.258 + 0.965i)23-s + (−0.866 − 0.499i)25-s − 1.41·29-s + (0.707 − 1.22i)31-s + (−0.366 + 1.36i)33-s + (0.500 + 0.866i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ -0.167 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479353376\)
\(L(\frac12)\) \(\approx\) \(1.479353376\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.576293761578498353837502769815, −8.822543943449985653933298039697, −7.73988844154243444718460885244, −7.44110102090776600317792327616, −6.20493028184626098195819348138, −5.41549075448988805583363110965, −4.22053640154578645373597622004, −3.87063040919870197470886857762, −3.14236150534354634902022405615, −1.68842444666166328293833989772, 1.17085640397907664339456941667, 1.78003672891434328012202813746, 2.97587656262498215359086644892, 4.21074757125071618393524401074, 5.05080093109892781243053355153, 6.06881414658068637079629875914, 6.68253212688072405575663466000, 7.61727483294266265295679010585, 8.261598991065441537781617036874, 8.943327346329701216300309995303

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