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Properties

Label 2-3024-9.7-c1-0-18
Degree $2$
Conductor $3024$
Sign $0.574 + 0.818i$
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  + (−1.87 − 3.24i)5-s + (0.5 − 0.866i)7-s + (1.82 − 3.16i)11-s + (2.77 + 4.80i)13-s + 7.20·17-s + 3.30·19-s + (2.49 + 4.32i)23-s + (−4.52 + 7.84i)25-s + (0.245 − 0.425i)29-s + (−1.94 − 3.37i)31-s − 3.74·35-s + 7.89·37-s + (2.38 + 4.12i)41-s + (−0.801 + 1.38i)43-s + (−4.81 + 8.34i)47-s + ⋯
L(s)  = 1  + (−0.838 − 1.45i)5-s + (0.188 − 0.327i)7-s + (0.550 − 0.953i)11-s + (0.769 + 1.33i)13-s + 1.74·17-s + 0.758·19-s + (0.520 + 0.900i)23-s + (−0.905 + 1.56i)25-s + (0.0455 − 0.0789i)29-s + (−0.349 − 0.605i)31-s − 0.633·35-s + 1.29·37-s + (0.372 + 0.644i)41-s + (−0.122 + 0.211i)43-s + (−0.703 + 1.21i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.931858911\)
\(L(\frac12)\) \(\approx\) \(1.931858911\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (1.87 + 3.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.82 + 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.77 - 4.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.20T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 + (-2.49 - 4.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.245 + 0.425i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.94 + 3.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + (-2.38 - 4.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.801 - 1.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.81 - 8.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.03T + 53T^{2} \)
59 \( 1 + (-0.754 - 1.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.70 - 2.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 + (-1.86 + 3.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.69 + 9.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.14T + 89T^{2} \)
97 \( 1 + (-5.45 + 9.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.599726469349775650259001226826, −7.891462906100643954972145950077, −7.39343018039540275612543451214, −6.18425175259014220968334092277, −5.51589366257725172726801777355, −4.61870116882289820286826992173, −3.91567944629780972464417871096, −3.26821934666052881492613411257, −1.38759150203442407588755738652, −0.919742027637884508891876815005, 0.973878351831064348344221583309, 2.47866885864658936300927670621, 3.31156080808254349093950594603, 3.81302409816764563686497307378, 5.05207410929542641090536232052, 5.84532713774263135296169489867, 6.72114073364657537116445747436, 7.40145234428364505549316540912, 7.896402715751944902271968171423, 8.674686083201488837702488264916

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