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Properties

Label 2-2400-8.5-c1-0-17
Degree $2$
Conductor $2400$
Sign $0.947 - 0.318i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  + i·3-s − 2.64·7-s − 9-s − 1.51i·11-s − 3.87i·13-s + 3.31·17-s + 7.08i·19-s − 2.64i·21-s − 4.82·23-s − i·27-s + 2.18i·29-s + 7.36·31-s + 1.51·33-s − 7.87i·37-s + 3.87·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.998·7-s − 0.333·9-s − 0.456i·11-s − 1.07i·13-s + 0.803·17-s + 1.62i·19-s − 0.576i·21-s − 1.00·23-s − 0.192i·27-s + 0.405i·29-s + 1.32·31-s + 0.263·33-s − 1.29i·37-s + 0.619·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451136222\)
\(L(\frac12)\) \(\approx\) \(1.451136222\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 + 1.51iT - 11T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 - 7.08iT - 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 2.18iT - 29T^{2} \)
31 \( 1 - 7.36T + 31T^{2} \)
37 \( 1 + 7.87iT - 37T^{2} \)
41 \( 1 - 8.72T + 41T^{2} \)
43 \( 1 - 1.01iT - 43T^{2} \)
47 \( 1 - 7.08T + 47T^{2} \)
53 \( 1 + 4.50iT - 53T^{2} \)
59 \( 1 + 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 - 1.01iT - 67T^{2} \)
71 \( 1 - 6.72T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−9.161021473544472458876143679823, −8.086511302344515980232351432067, −7.76942306960994776842173358833, −6.42673546360306079418320552004, −5.88314782351001109771632358779, −5.20617190738389387383026705024, −3.86966211653493866683065667606, −3.46838988394947283658350777304, −2.43128340145661761596844591996, −0.74398117371210092262737769271, 0.77879348686956699590168710429, 2.17599943435635160806574549861, 2.97894481934235703887237635005, 4.09126826011184126008985000217, 4.92944873048515401704111303829, 6.12636039955725407700081250723, 6.54735641707656188606800181013, 7.31192131440921268728700779226, 8.055143406604106310796091804194, 9.045157984358803352914362327837

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