Properties

Label 2-3024-63.58-c1-0-37
Degree $2$
Conductor $3024$
Sign $-0.437 + 0.899i$
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  + (1.84 + 3.20i)5-s + (−2.64 + 0.0963i)7-s + (0.738 − 1.27i)11-s + (−1.34 + 2.33i)13-s + (−3.28 − 5.69i)17-s + (0.444 − 0.769i)19-s + (−3.14 − 5.44i)23-s + (−4.34 + 7.52i)25-s + (−1.25 − 2.17i)29-s − 6.81·31-s + (−5.19 − 8.29i)35-s + (−1.38 + 2.40i)37-s + (2.05 − 3.56i)41-s + (−0.00618 − 0.0107i)43-s − 6.98·47-s + ⋯
L(s)  = 1  + (0.827 + 1.43i)5-s + (−0.999 + 0.0364i)7-s + (0.222 − 0.385i)11-s + (−0.374 + 0.648i)13-s + (−0.797 − 1.38i)17-s + (0.101 − 0.176i)19-s + (−0.655 − 1.13i)23-s + (−0.868 + 1.50i)25-s + (−0.233 − 0.403i)29-s − 1.22·31-s + (−0.878 − 1.40i)35-s + (−0.228 + 0.395i)37-s + (0.321 − 0.556i)41-s + (−0.000943 − 0.00163i)43-s − 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3877825161\)
\(L(\frac12)\) \(\approx\) \(0.3877825161\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0963i)T \)
good5 \( 1 + (-1.84 - 3.20i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.738 + 1.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.34 - 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.28 + 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.14 + 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.05 + 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.00618 + 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.98T + 47T^{2} \)
53 \( 1 + (-1.60 - 2.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.90T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 - 9.46T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (6.03 + 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + (-2.23 - 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.43 + 7.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.58 + 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.697140325957357065623071068739, −7.34752142045281446608391007762, −6.92599483582059105724975808363, −6.32615934929614903885475011616, −5.68976779071702963563867155416, −4.53629784977037140641364382322, −3.47967061857905048601740424727, −2.71304172451463075834861355065, −2.07021849800184150426401204094, −0.11277964898076659454354851589, 1.35946584386771098146239441733, 2.17811408401637202763882551329, 3.52711303888128697470743459351, 4.27214411463124304078244557298, 5.32104996155651387047909722982, 5.77379056150765602425670845817, 6.56506709757902784459731705277, 7.51259178377193587438799119911, 8.407341530483822749792684660775, 9.010744654633529043962618200591

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