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Properties

Label 2-3024-21.2-c0-0-1
Degree $2$
Conductor $3024$
Sign $0.832 - 0.553i$
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)7-s + 2·13-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)31-s + (−1 + 1.73i)37-s + 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)61-s + (1 + 1.73i)67-s + (0.5 + 0.866i)73-s + (1 − 1.73i)79-s + (1 + 1.73i)91-s − 97-s + (1 − 1.73i)103-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + 2·13-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)31-s + (−1 + 1.73i)37-s + 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)61-s + (1 + 1.73i)67-s + (0.5 + 0.866i)73-s + (1 − 1.73i)79-s + (1 + 1.73i)91-s − 97-s + (1 − 1.73i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.403825573\)
\(L(\frac12)\) \(\approx\) \(1.403825573\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - 2T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
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

−8.636490401739131474921612612021, −8.470877040860213199680862253172, −7.67341998901374022765975351397, −6.43994571219116060256033335697, −6.03357412953258267668983010155, −5.25370995451373661846185443757, −4.19258181092855002355422033845, −3.49107410634711396724391598675, −2.30514510687235254119250570635, −1.37554072832928014524620016654, 1.02668165045805675640069673409, 2.05408685603753019419363762605, 3.52997627047824460349164794222, 3.92651313435897278522217883641, 4.98411873408828019884347305389, 5.78913769048769684396343853399, 6.65823201422331666216501939357, 7.30878386537984907077780515697, 8.128001158619050544783132378622, 8.815271575623316302579779526192

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