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Properties

Label 2-3024-252.151-c0-0-1
Degree $2$
Conductor $3024$
Sign $0.235 - 0.971i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + i·7-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)29-s + (−0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s − 49-s + (−0.5 + 0.866i)53-s + 0.999i·55-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + i·7-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)29-s + (−0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s − 49-s + (−0.5 + 0.866i)53-s + 0.999i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190116992\)
\(L(\frac12)\) \(\approx\) \(1.190116992\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.167442716959600473473819437665, −8.348261919823970968680735445138, −7.51084352074585991551063238069, −6.75268932907824968750898913303, −6.17288967725007664859722712857, −5.30163200589549857175929041222, −4.30926189742245819454476845363, −3.33404208974456115862302757411, −2.76110221533156368073214882061, −1.44414916526342576961560918446, 0.831670549984707771698456759737, 1.79637088202513473231558827874, 3.55517409579553639299482338461, 3.87019715689865681765213299105, 4.76757487236246300674426771469, 5.69914134465009183984767388286, 6.43856211728680262564042335087, 7.53460511159238040222676165358, 7.900563936956519697183636447426, 8.601639691440636010198165021391

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