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Properties

Label 2-3024-336.317-c0-0-3
Degree $2$
Conductor $3024$
Sign $0.657 + 0.753i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.866 + 0.5i)7-s + (0.707 − 0.707i)8-s + (0.258 − 0.965i)11-s + (−1 − i)13-s + (0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.36 + 0.366i)19-s − i·22-s + (0.707 − 1.22i)23-s + (−0.866 + 0.5i)25-s + (−1.22 − 0.707i)26-s + 28-s + (0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.866 + 0.5i)7-s + (0.707 − 0.707i)8-s + (0.258 − 0.965i)11-s + (−1 − i)13-s + (0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.36 + 0.366i)19-s − i·22-s + (0.707 − 1.22i)23-s + (−0.866 + 0.5i)25-s + (−1.22 − 0.707i)26-s + 28-s + (0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.657 + 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.453761771\)
\(L(\frac12)\) \(\approx\) \(2.453761771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.634685319387344620441098949171, −8.047836429088433965123138694129, −7.24322888704330665352282759311, −6.12893502504868718765192818905, −5.83628935736972616241901639217, −4.72614342664059225830361502397, −4.36920106358002948077397575687, −3.00407990482783671564491892803, −2.51167670967082394809197664434, −1.22237982624358429343895283768, 1.78270369136478985852233005520, 2.37225704890988395031395775841, 3.76356645285759260481072313779, 4.55065204423146444500232858957, 4.80806704272720986052914759991, 5.95799361808473824725834605279, 6.77823123581915663781771628105, 7.39317704055778451846484188903, 7.903939972959127474212549913070, 8.952610882279069082085985156031

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