Properties

Label 2-2240-280.139-c1-0-8
Degree $2$
Conductor $2240$
Sign $-0.258 - 0.965i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  − 1.17·3-s − 2.23i·5-s + 2.64i·7-s − 1.62·9-s + 0.359·11-s − 5.64i·13-s + 2.62i·15-s − 2.03·17-s − 3.10i·21-s − 5.00·25-s + 5.42·27-s + 10.7i·29-s − 0.422·33-s + 5.91·35-s + 6.62i·39-s + ⋯
L(s)  = 1  − 0.677·3-s − 0.999i·5-s + 0.999i·7-s − 0.541·9-s + 0.108·11-s − 1.56i·13-s + 0.677i·15-s − 0.492·17-s − 0.677i·21-s − 1.00·25-s + 1.04·27-s + 1.99i·29-s − 0.0735·33-s + 0.999·35-s + 1.06i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4596376619\)
\(L(\frac12)\) \(\approx\) \(0.4596376619\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 2.23iT \)
7 \( 1 - 2.64iT \)
good3 \( 1 + 1.17T + 3T^{2} \)
11 \( 1 - 0.359T + 11T^{2} \)
13 \( 1 + 5.64iT - 13T^{2} \)
17 \( 1 + 2.03T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10.7iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 5.71iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−8.994561302008714846853602828352, −8.637285257364019144569340645944, −7.921685277167609218077318605658, −6.79214568321244172748752521865, −5.81121358856903259652964970675, −5.38899895476033934153175447645, −4.81523851196650237964420666202, −3.47490104137960477615137754542, −2.48722158026240220387370487902, −1.10589186381375166421641249296, 0.19210587454227862787209171121, 1.83915840068172775813795217436, 2.95390311806928971607570804087, 4.06080367629849997175687530690, 4.62534658954236887062585388545, 5.98350152833076953738796502535, 6.40539141730932296941036028520, 7.12948540945695218419323818546, 7.83346140708982945201594647300, 8.883214841301674045355279734853

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