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Properties

Label 2-3024-63.47-c1-0-3
Degree $2$
Conductor $3024$
Sign $-0.118 - 0.992i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  − 2.74·5-s + (−1.70 − 2.02i)7-s − 0.418i·11-s + (−1.32 + 0.765i)13-s + (−1.95 − 3.38i)17-s + (5.11 + 2.95i)19-s − 8.92i·23-s + 2.52·25-s + (−6.00 − 3.46i)29-s + (3.05 + 1.76i)31-s + (4.67 + 5.55i)35-s + (−4.54 + 7.87i)37-s + (1.06 + 1.84i)41-s + (5.77 − 10.0i)43-s + (−0.885 − 1.53i)47-s + ⋯
L(s)  = 1  − 1.22·5-s + (−0.644 − 0.764i)7-s − 0.126i·11-s + (−0.367 + 0.212i)13-s + (−0.473 − 0.820i)17-s + (1.17 + 0.678i)19-s − 1.86i·23-s + 0.505·25-s + (−1.11 − 0.643i)29-s + (0.548 + 0.316i)31-s + (0.790 + 0.938i)35-s + (−0.747 + 1.29i)37-s + (0.165 + 0.287i)41-s + (0.881 − 1.52i)43-s + (−0.129 − 0.223i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3748883172\)
\(L(\frac12)\) \(\approx\) \(0.3748883172\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 2.02i)T \)
good5 \( 1 + 2.74T + 5T^{2} \)
11 \( 1 + 0.418iT - 11T^{2} \)
13 \( 1 + (1.32 - 0.765i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.11 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.92iT - 23T^{2} \)
29 \( 1 + (6.00 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.05 - 1.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.54 - 7.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.77 + 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.885 + 1.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.39 - 1.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.02 - 3.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.61 - 0.932i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.38 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (-1.65 + 0.952i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.433 + 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.45 - 5.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.88 - 8.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.200 + 0.115i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.838031088676150501414177428942, −8.090988364642787321075039265678, −7.32661805808054635513694154257, −6.94193957397175342995384979997, −5.97759545098757159171167399601, −4.85369168947897092099626866536, −4.16684549973001851724649204010, −3.46069176222451646535312150746, −2.55984297912378260131979470332, −0.886653697735440450131945832408, 0.15276506927490245422122096351, 1.78857250953251104173656834278, 3.09391988810625900982275398419, 3.58202651317125664254056162162, 4.59131534430647262883913315229, 5.48507356360226715058630054899, 6.18608166009781388959553313988, 7.39983638011094251143536803909, 7.50547992707454397716849671394, 8.525229463136706971606867419042

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