Properties

Label 2-3024-21.20-c1-0-53
Degree $2$
Conductor $3024$
Sign $-0.871 + 0.489i$
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  − 2.85·5-s + (2.30 − 1.29i)7-s − 3.55i·11-s − 2.43i·13-s + 0.860·17-s + 3.55i·19-s − 5.94i·23-s + 3.15·25-s − 2.88i·29-s + 9.01i·31-s + (−6.58 + 3.70i)35-s + 7.43·37-s − 6.85·41-s − 3.48·43-s − 0.263·47-s + ⋯
L(s)  = 1  − 1.27·5-s + (0.871 − 0.489i)7-s − 1.07i·11-s − 0.675i·13-s + 0.208·17-s + 0.816i·19-s − 1.23i·23-s + 0.631·25-s − 0.535i·29-s + 1.61i·31-s + (−1.11 + 0.625i)35-s + 1.22·37-s − 1.07·41-s − 0.532·43-s − 0.0384·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.871 + 0.489i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.871 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7725576733\)
\(L(\frac12)\) \(\approx\) \(0.7725576733\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 + 1.29i)T \)
good5 \( 1 + 2.85T + 5T^{2} \)
11 \( 1 + 3.55iT - 11T^{2} \)
13 \( 1 + 2.43iT - 13T^{2} \)
17 \( 1 - 0.860T + 17T^{2} \)
19 \( 1 - 3.55iT - 19T^{2} \)
23 \( 1 + 5.94iT - 23T^{2} \)
29 \( 1 + 2.88iT - 29T^{2} \)
31 \( 1 - 9.01iT - 31T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 + 6.85T + 41T^{2} \)
43 \( 1 + 3.48T + 43T^{2} \)
47 \( 1 + 0.263T + 47T^{2} \)
53 \( 1 + 7.76iT - 53T^{2} \)
59 \( 1 + 9.72T + 59T^{2} \)
61 \( 1 - 1.62iT - 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 - 15.1iT - 71T^{2} \)
73 \( 1 + 9.20iT - 73T^{2} \)
79 \( 1 - 7.55T + 79T^{2} \)
83 \( 1 - 0.922T + 83T^{2} \)
89 \( 1 - 4.90T + 89T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−8.170772839733194170122054265498, −7.951230938543782061039624711721, −7.05333376730630897222999630522, −6.15799835016965956280435583522, −5.20435529976040831216737486027, −4.42532756766906986049857251452, −3.66268243766790417020687880897, −2.91397596105138745819090521083, −1.36756641824228951354676192625, −0.26082198968565517885918387187, 1.43372201411854651245348145693, 2.46485156927664489150835680568, 3.62100341766712604190865273965, 4.46397596824156485774456476069, 4.91849414803177812266262265216, 5.99695410937653094758699616313, 7.02195767847487670155268859915, 7.68207403968966488269281764308, 8.026306658688862188244855806434, 9.121278150063967814116562079651

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