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Properties

Label 2-4560-76.75-c1-0-71
Degree $2$
Conductor $4560$
Sign $-0.0722 + 0.997i$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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  + 3-s + 5-s − 4.54i·7-s + 9-s + 2.47i·11-s − 2.88i·13-s + 15-s + 6.99·17-s + (−3.60 − 2.44i)19-s − 4.54i·21-s − 2.99i·23-s + 25-s + 27-s + 0.990i·29-s − 3.62·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.71i·7-s + 0.333·9-s + 0.745i·11-s − 0.800i·13-s + 0.258·15-s + 1.69·17-s + (−0.827 − 0.561i)19-s − 0.991i·21-s − 0.624i·23-s + 0.200·25-s + 0.192·27-s + 0.183i·29-s − 0.651·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0722 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0722 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0722 + 0.997i$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ -0.0722 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.497509104\)
\(L(\frac12)\) \(\approx\) \(2.497509104\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 + (3.60 + 2.44i)T \)
good7 \( 1 + 4.54iT - 7T^{2} \)
11 \( 1 - 2.47iT - 11T^{2} \)
13 \( 1 + 2.88iT - 13T^{2} \)
17 \( 1 - 6.99T + 17T^{2} \)
23 \( 1 + 2.99iT - 23T^{2} \)
29 \( 1 - 0.990iT - 29T^{2} \)
31 \( 1 + 3.62T + 31T^{2} \)
37 \( 1 + 9.46iT - 37T^{2} \)
41 \( 1 - 1.54iT - 41T^{2} \)
43 \( 1 - 7.98iT - 43T^{2} \)
47 \( 1 + 0.413iT - 47T^{2} \)
53 \( 1 + 6.45iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 6.65T + 61T^{2} \)
67 \( 1 + 7.16T + 67T^{2} \)
71 \( 1 + 8.62T + 71T^{2} \)
73 \( 1 - 0.592T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 3.02iT - 83T^{2} \)
89 \( 1 - 18.4iT - 89T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−8.032680274861063155305958001974, −7.35642358618115722803828297470, −6.99797389750262662859879276225, −5.98068654680942571573577993475, −5.07956991686497339183783879857, −4.25831320596510254836951215153, −3.60566085607083160136688653091, −2.71474598292838391376497416172, −1.60719624337733606535935343754, −0.63143365513192384401251120722, 1.43003570956903891449096492048, 2.23005330504443654257071007524, 3.04880528147189655397336899029, 3.78603833447855785073281603884, 4.97173726562427018868292521350, 5.75291760389403964333398479217, 6.06740940712024013501173597032, 7.11928001615065846654626726900, 7.980304603069036213598289970254, 8.763287928386404284854326688675

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