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Properties

Label 2-3024-63.25-c1-0-31
Degree $2$
Conductor $3024$
Sign $0.462 + 0.886i$
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  + (2.07 − 3.58i)5-s + (−2.19 + 1.48i)7-s + (−0.434 − 0.752i)11-s + (2.86 + 4.95i)13-s + (1.44 − 2.50i)17-s + (2.00 + 3.47i)19-s + (2.91 − 5.04i)23-s + (−6.09 − 10.5i)25-s + (0.900 − 1.55i)29-s + 2.96·31-s + (0.789 + 10.9i)35-s + (−2.64 − 4.58i)37-s + (5.89 + 10.2i)41-s + (2.00 − 3.47i)43-s + 2.34·47-s + ⋯
L(s)  = 1  + (0.926 − 1.60i)5-s + (−0.827 + 0.561i)7-s + (−0.130 − 0.226i)11-s + (0.793 + 1.37i)13-s + (0.350 − 0.607i)17-s + (0.460 + 0.797i)19-s + (0.607 − 1.05i)23-s + (−1.21 − 2.11i)25-s + (0.167 − 0.289i)29-s + 0.531·31-s + (0.133 + 1.84i)35-s + (−0.435 − 0.754i)37-s + (0.920 + 1.59i)41-s + (0.306 − 0.530i)43-s + 0.341·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.462 + 0.886i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.462 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.085286075\)
\(L(\frac12)\) \(\approx\) \(2.085286075\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 1.48i)T \)
good5 \( 1 + (-2.07 + 3.58i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.434 + 0.752i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.86 - 4.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.44 + 2.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 - 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.91 + 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.900 + 1.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + (2.64 + 4.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.89 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.00 + 3.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.34T + 47T^{2} \)
53 \( 1 + (-1.09 + 1.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.05T + 59T^{2} \)
61 \( 1 - 5.63T + 61T^{2} \)
67 \( 1 - 2.51T + 67T^{2} \)
71 \( 1 + 1.09T + 71T^{2} \)
73 \( 1 + (-0.723 + 1.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.12T + 79T^{2} \)
83 \( 1 + (-2.18 + 3.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.83 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.98 - 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.709475403205754628601435543444, −8.172709752524934269995646737651, −6.90721944809883552666036873444, −6.11992444628243660992024761998, −5.63503647164046666716957968902, −4.77357414959955223885799434646, −4.01349765829263782478424947861, −2.77775356762537130410335084203, −1.77001930015756686888652142719, −0.76443006208061171198409506246, 1.10993047245377566872401595585, 2.52948103743496200122152675865, 3.17928071942590431252757876133, 3.77314242157514983289665739107, 5.30925325625260844937843966156, 5.90593903475832977147223587394, 6.61669102952469228999165751888, 7.22150937192847046963413894644, 7.87308596501458826899192538299, 9.047330335458629299101835805153

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