Properties

Label 2-6048-12.11-c1-0-38
Degree $2$
Conductor $6048$
Sign $0.707 - 0.707i$
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  + 0.680i·5-s + i·7-s − 1.70·11-s + 5.59·13-s − 3.68i·17-s + 4.46i·19-s − 1.16·23-s + 4.53·25-s + 1.77i·29-s − 2.58i·31-s − 0.680·35-s + 1.36·37-s + 1.31i·41-s + 4.71i·43-s + 3.87·47-s + ⋯
L(s)  = 1  + 0.304i·5-s + 0.377i·7-s − 0.513·11-s + 1.55·13-s − 0.894i·17-s + 1.02i·19-s − 0.242·23-s + 0.907·25-s + 0.329i·29-s − 0.464i·31-s − 0.115·35-s + 0.223·37-s + 0.205i·41-s + 0.718i·43-s + 0.565·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.028667621\)
\(L(\frac12)\) \(\approx\) \(2.028667621\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.680iT - 5T^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 + 3.68iT - 17T^{2} \)
19 \( 1 - 4.46iT - 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 - 1.77iT - 29T^{2} \)
31 \( 1 + 2.58iT - 31T^{2} \)
37 \( 1 - 1.36T + 37T^{2} \)
41 \( 1 - 1.31iT - 41T^{2} \)
43 \( 1 - 4.71iT - 43T^{2} \)
47 \( 1 - 3.87T + 47T^{2} \)
53 \( 1 + 6.39iT - 53T^{2} \)
59 \( 1 - 0.948T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 + 1.28iT - 67T^{2} \)
71 \( 1 - 3.09T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 - 1.72iT - 79T^{2} \)
83 \( 1 - 7.69T + 83T^{2} \)
89 \( 1 + 0.962iT - 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.171982718320611146971175415274, −7.53210609357689987492261295121, −6.65466125800075259619059265532, −6.04921055341478705961194598676, −5.40289550755453513517518778578, −4.55039638349690451803034052650, −3.61243715586064724818549675784, −2.97620546487330044795037745849, −2.00023693758714212507218213439, −0.910430080580916538077047288434, 0.66327007916709642747725611270, 1.59676175212266313027200696047, 2.71782396818447004851299855948, 3.61626361889285964598719170914, 4.29483073257915870848780039507, 5.10203024039185233904085498511, 5.90717042563767409064823284389, 6.51994234714341042483467150217, 7.29681922056566914484813764161, 8.063329238246267115351513967435

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