Properties

Label 2-6048-8.5-c1-0-77
Degree $2$
Conductor $6048$
Sign $0.534 + 0.845i$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
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  + 3.16i·5-s + 7-s − 5.44i·11-s + 3.61i·13-s + 3.27·17-s − 3.20i·19-s − 0.673·23-s − 5.04·25-s − 2.85i·29-s − 3.71·31-s + 3.16i·35-s − 11.9i·37-s − 7.44·41-s − 12.5i·43-s + 4.06·47-s + ⋯
L(s)  = 1  + 1.41i·5-s + 0.377·7-s − 1.64i·11-s + 1.00i·13-s + 0.794·17-s − 0.735i·19-s − 0.140·23-s − 1.00·25-s − 0.529i·29-s − 0.667·31-s + 0.535i·35-s − 1.97i·37-s − 1.16·41-s − 1.91i·43-s + 0.592·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ 0.534 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.525942373\)
\(L(\frac12)\) \(\approx\) \(1.525942373\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 - 3.16iT - 5T^{2} \)
11 \( 1 + 5.44iT - 11T^{2} \)
13 \( 1 - 3.61iT - 13T^{2} \)
17 \( 1 - 3.27T + 17T^{2} \)
19 \( 1 + 3.20iT - 19T^{2} \)
23 \( 1 + 0.673T + 23T^{2} \)
29 \( 1 + 2.85iT - 29T^{2} \)
31 \( 1 + 3.71T + 31T^{2} \)
37 \( 1 + 11.9iT - 37T^{2} \)
41 \( 1 + 7.44T + 41T^{2} \)
43 \( 1 + 12.5iT - 43T^{2} \)
47 \( 1 - 4.06T + 47T^{2} \)
53 \( 1 + 0.291iT - 53T^{2} \)
59 \( 1 - 0.0209iT - 59T^{2} \)
61 \( 1 + 5.34iT - 61T^{2} \)
67 \( 1 - 6.20iT - 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 1.35T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 13.6iT - 83T^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 - 9.60T + 97T^{2} \)
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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

−7.80284207360687434256114378521, −7.22169466933446873113013496619, −6.60673782674577104649775892287, −5.86768792973535299591023892578, −5.29871701385687679942346659620, −4.04352197871843801652988460482, −3.49809585644840302665824117977, −2.70359541068741930922731850300, −1.85131347334268533912870648477, −0.40227707903944276014061178720, 1.16199786682216951785504856091, 1.68329285250533953858609797135, 2.92552824878808738128140229109, 3.94110098616831913401292048524, 4.80011685095517154162436358429, 5.08637910644176595605978862474, 5.88005653621634258726932712870, 6.84647152245267091504138855503, 7.80959273678547931839015471176, 8.001770320026129584516770520742

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