Properties

Label 2-6048-24.11-c1-0-82
Degree $2$
Conductor $6048$
Sign $-0.917 + 0.396i$
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  − 0.530·5-s + i·7-s − 0.905i·11-s − 4.38i·13-s − 1.25i·17-s − 0.00364·19-s − 0.778·23-s − 4.71·25-s + 5.43·29-s + 3.49i·31-s − 0.530i·35-s − 5.34i·37-s + 1.74i·41-s + 0.259·43-s + 3.75·47-s + ⋯
L(s)  = 1  − 0.237·5-s + 0.377i·7-s − 0.272i·11-s − 1.21i·13-s − 0.304i·17-s − 0.000837·19-s − 0.162·23-s − 0.943·25-s + 1.00·29-s + 0.628i·31-s − 0.0895i·35-s − 0.878i·37-s + 0.272i·41-s + 0.0395·43-s + 0.548·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.917 + 0.396i$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ -0.917 + 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5078945075\)
\(L(\frac12)\) \(\approx\) \(0.5078945075\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 + 0.530T + 5T^{2} \)
11 \( 1 + 0.905iT - 11T^{2} \)
13 \( 1 + 4.38iT - 13T^{2} \)
17 \( 1 + 1.25iT - 17T^{2} \)
19 \( 1 + 0.00364T + 19T^{2} \)
23 \( 1 + 0.778T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 - 3.49iT - 31T^{2} \)
37 \( 1 + 5.34iT - 37T^{2} \)
41 \( 1 - 1.74iT - 41T^{2} \)
43 \( 1 - 0.259T + 43T^{2} \)
47 \( 1 - 3.75T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 - 5.73iT - 59T^{2} \)
61 \( 1 - 0.555iT - 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + 1.96iT - 79T^{2} \)
83 \( 1 + 5.82iT - 83T^{2} \)
89 \( 1 + 1.94iT - 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−7.66180824007601582782940512820, −7.28510511715512386113692443670, −6.14069831943445985433205796346, −5.74447490777372780993664565599, −4.90809625334516171921343225051, −4.09465335906014947334258718164, −3.16788032200858322029506373491, −2.56031141544875883552891265246, −1.32146007378122249643601002097, −0.13230541179132730607373874619, 1.30567106949656293449875793487, 2.19777440012886365387940156176, 3.21564224636961907834829065413, 4.22991890641166154426263118480, 4.46337132277831322069424058748, 5.60214854996155587308609467201, 6.32895152578841529058316440838, 6.99964218793579890938787117647, 7.62657007105792623940890218194, 8.359501133922020662618300794079

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