Properties

Label 2-6048-24.11-c1-0-18
Degree $2$
Conductor $6048$
Sign $-0.525 - 0.850i$
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  − 3.69·5-s + i·7-s + 2.05i·11-s + 4.65i·13-s − 5.73i·17-s − 3.27·19-s + 4.45·23-s + 8.66·25-s + 10.2·29-s + 1.26i·31-s − 3.69i·35-s + 3.25i·37-s − 8.72i·41-s + 2.56·43-s − 2.03·47-s + ⋯
L(s)  = 1  − 1.65·5-s + 0.377i·7-s + 0.620i·11-s + 1.28i·13-s − 1.39i·17-s − 0.750·19-s + 0.928·23-s + 1.73·25-s + 1.90·29-s + 0.226i·31-s − 0.624i·35-s + 0.535i·37-s − 1.36i·41-s + 0.391·43-s − 0.296·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.525 - 0.850i$
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: \((2,\ 6048,\ (\ :1/2),\ -0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8204557827\)
\(L(\frac12)\) \(\approx\) \(0.8204557827\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 - 2.05iT - 11T^{2} \)
13 \( 1 - 4.65iT - 13T^{2} \)
17 \( 1 + 5.73iT - 17T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 - 3.25iT - 37T^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 - 2.56T + 43T^{2} \)
47 \( 1 + 2.03T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 0.851iT - 59T^{2} \)
61 \( 1 + 2.31iT - 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 2.35iT - 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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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

−8.446562222063016332159320560660, −7.43886417693810772717746213305, −7.03852343342542467335441118922, −6.48341187756692198842669877208, −5.18689831201180103546500871926, −4.56405042983457474469860393084, −4.09318463725837068408896768669, −3.09362618660167135851594956997, −2.33060721583239821541220918725, −0.947413416515852671888775743825, 0.28877515235950619382355533053, 1.17969773418757892848731493822, 2.82082134016050400479320780172, 3.36484025182630661820506865074, 4.18497273482775981634034500543, 4.69171675674302366159135560834, 5.76956594668853604522838814318, 6.46401550406769119579729445820, 7.30154482704824784448554298997, 7.918609120395105949908595455581

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