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Properties

Label 2-6048-24.11-c1-0-40
Degree $2$
Conductor $6048$
Sign $0.359 - 0.933i$
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  + 2.29·5-s + i·7-s + 4.02i·11-s − 1.82i·13-s − 0.430i·17-s − 5.01·19-s + 3.49·23-s + 0.274·25-s + 2.16·29-s + 2.10i·31-s + 2.29i·35-s − 2.19i·37-s − 4.35i·41-s + 12.0·43-s − 1.72·47-s + ⋯
L(s)  = 1  + 1.02·5-s + 0.377i·7-s + 1.21i·11-s − 0.506i·13-s − 0.104i·17-s − 1.14·19-s + 0.728·23-s + 0.0548·25-s + 0.401·29-s + 0.378i·31-s + 0.388i·35-s − 0.361i·37-s − 0.679i·41-s + 1.83·43-s − 0.251·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.359 - 0.933i$
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: \((2,\ 6048,\ (\ :1/2),\ 0.359 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.210684709\)
\(L(\frac12)\) \(\approx\) \(2.210684709\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.29T + 5T^{2} \)
11 \( 1 - 4.02iT - 11T^{2} \)
13 \( 1 + 1.82iT - 13T^{2} \)
17 \( 1 + 0.430iT - 17T^{2} \)
19 \( 1 + 5.01T + 19T^{2} \)
23 \( 1 - 3.49T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 - 2.10iT - 31T^{2} \)
37 \( 1 + 2.19iT - 37T^{2} \)
41 \( 1 + 4.35iT - 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + 1.72T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 - 8.44iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 - 15.2iT - 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 - 7.44iT - 89T^{2} \)
97 \( 1 - 1.68T + 97T^{2} \)
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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

−8.327410539560169470770881820812, −7.32183976789085208321285954549, −6.86998649735055059029633889530, −5.93177582648404346705724466295, −5.49747793842201174750875438849, −4.64307905650750256882308046233, −3.90508238686934805540216517334, −2.57509222336722035136252982175, −2.24124328736547502653919069310, −1.11274547111430939458253153087, 0.58484955435015787041544301465, 1.69471227447157148022727772988, 2.53803171059597854763468702386, 3.44608539550108009979942687632, 4.32168206281954867137145645023, 5.10141998167936091623921234095, 6.09190781175015727271974683909, 6.23108631316597668148255841386, 7.15923014764845413631481164142, 8.003634158013348498587980289262

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