Properties

Degree $2$
Conductor $3024$
Sign $-0.307 + 0.951i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.240 − 0.416i)5-s + (1.92 − 1.81i)7-s + (−1.69 + 2.92i)11-s + (−2.86 + 4.95i)13-s + (−2.75 − 4.77i)17-s + (−2.18 + 3.77i)19-s + (−1.81 − 3.14i)23-s + (2.38 − 4.12i)25-s + (−1.53 − 2.65i)29-s + 9.34·31-s + (−1.21 − 0.367i)35-s + (1.48 − 2.57i)37-s + (6.29 − 10.9i)41-s + (−1.90 − 3.30i)43-s − 3.76·47-s + ⋯
L(s)  = 1  + (−0.107 − 0.186i)5-s + (0.728 − 0.684i)7-s + (−0.509 + 0.882i)11-s + (−0.793 + 1.37i)13-s + (−0.668 − 1.15i)17-s + (−0.500 + 0.866i)19-s + (−0.378 − 0.654i)23-s + (0.476 − 0.825i)25-s + (−0.284 − 0.492i)29-s + 1.67·31-s + (−0.206 − 0.0621i)35-s + (0.244 − 0.422i)37-s + (0.983 − 1.70i)41-s + (−0.291 − 0.504i)43-s − 0.549·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.307 + 0.951i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.307 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.094822257\)
\(L(\frac12)\) \(\approx\) \(1.094822257\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.92 + 1.81i)T \)
good5 \( 1 + (0.240 + 0.416i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.69 - 2.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.86 - 4.95i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.75 + 4.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.18 - 3.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.81 + 3.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.53 + 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.34T + 31T^{2} \)
37 \( 1 + (-1.48 + 2.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.29 + 10.9i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.76T + 47T^{2} \)
53 \( 1 + (5.57 + 9.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.42T + 59T^{2} \)
61 \( 1 + 7.28T + 61T^{2} \)
67 \( 1 + 2.57T + 67T^{2} \)
71 \( 1 + 3.94T + 71T^{2} \)
73 \( 1 + (0.862 + 1.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.59T + 79T^{2} \)
83 \( 1 + (0.119 + 0.206i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.648 - 1.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.02 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.419415906886660568308117978289, −7.72329181161483160447027764570, −7.02952191132091208954924785952, −6.44579687591298253354915323355, −5.16517717379828648206912045327, −4.47999440502952103655323460571, −4.13904825657711726099035190002, −2.51077784210994807658799885833, −1.89442768315964702994696705796, −0.34637143951640553573530008123, 1.24655627297461379206143541317, 2.59627590773679715012067062814, 3.09043088977034278327550988653, 4.42498918068309836796446032467, 5.11787256043310498622046345537, 5.87724429482712917199349254268, 6.56424629980673940931750223248, 7.81056193384943124881999450146, 8.039793747566969404926340325362, 8.814385688213941051481158784224

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