Properties

Label 2-6048-8.5-c1-0-81
Degree $2$
Conductor $6048$
Sign $-0.955 + 0.295i$
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  − 3.04i·5-s − 7-s + 0.128i·11-s + 6.30i·13-s + 5.32·17-s + 6.68i·19-s − 5.18·23-s − 4.26·25-s − 9.96i·29-s − 3.27·31-s + 3.04i·35-s − 0.796i·37-s − 2.96·41-s − 6.99i·43-s − 4.76·47-s + ⋯
L(s)  = 1  − 1.36i·5-s − 0.377·7-s + 0.0387i·11-s + 1.74i·13-s + 1.29·17-s + 1.53i·19-s − 1.08·23-s − 0.852·25-s − 1.85i·29-s − 0.587·31-s + 0.514i·35-s − 0.131i·37-s − 0.463·41-s − 1.06i·43-s − 0.695·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.955 + 0.295i$
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: \((2,\ 6048,\ (\ :1/2),\ -0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6363884912\)
\(L(\frac12)\) \(\approx\) \(0.6363884912\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.04iT - 5T^{2} \)
11 \( 1 - 0.128iT - 11T^{2} \)
13 \( 1 - 6.30iT - 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 - 6.68iT - 19T^{2} \)
23 \( 1 + 5.18T + 23T^{2} \)
29 \( 1 + 9.96iT - 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 + 0.796iT - 37T^{2} \)
41 \( 1 + 2.96T + 41T^{2} \)
43 \( 1 + 6.99iT - 43T^{2} \)
47 \( 1 + 4.76T + 47T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + 14.6iT - 61T^{2} \)
67 \( 1 - 1.05iT - 67T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 - 6.46iT - 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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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

−7.985171488897973483780745965028, −7.08775815979460705932286695355, −6.14352382034057490290990421258, −5.69504964064351602144460072156, −4.80591948104383450471270261133, −4.07088398213496117407664420317, −3.55523944292039785373435042107, −2.04454010798360477649298658347, −1.48525473096577973706151091988, −0.16472322014360683173332056853, 1.21109595020964426405112146023, 2.69615424271880053651900226617, 3.03455951428175489244179491749, 3.69869899148043290602319778187, 4.92599643154723222860600478546, 5.66227919834459264111869571897, 6.25279087281403197588905787413, 7.16530675738397937874874698741, 7.44297268923464151392858880437, 8.289451056125251814807567253710

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