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Properties

Label 2-3024-252.11-c1-0-27
Degree $2$
Conductor $3024$
Sign $0.998 + 0.0477i$
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  + (1.5 + 0.866i)5-s + (2 − 1.73i)7-s + (1.5 + 2.59i)11-s + (−0.5 − 0.866i)13-s + (−1.5 − 0.866i)17-s + (1.5 − 0.866i)19-s + (1.5 − 2.59i)23-s + (−1 − 1.73i)25-s + (7.5 + 4.33i)29-s + 3.46i·31-s + (4.5 − 0.866i)35-s + (−2.5 − 4.33i)37-s + (4.5 − 2.59i)41-s + (10.5 + 6.06i)43-s + (1.00 − 6.92i)49-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + (0.755 − 0.654i)7-s + (0.452 + 0.783i)11-s + (−0.138 − 0.240i)13-s + (−0.363 − 0.210i)17-s + (0.344 − 0.198i)19-s + (0.312 − 0.541i)23-s + (−0.200 − 0.346i)25-s + (1.39 + 0.804i)29-s + 0.622i·31-s + (0.760 − 0.146i)35-s + (−0.410 − 0.711i)37-s + (0.702 − 0.405i)41-s + (1.60 + 0.924i)43-s + (0.142 − 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.998 + 0.0477i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.998 + 0.0477i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462875828\)
\(L(\frac12)\) \(\approx\) \(2.462875828\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.5 - 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.5 - 6.06i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (4.5 + 2.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (7.5 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.745578407059887458438492819745, −7.915188147452684452951377043894, −7.08349400963915650626717242356, −6.65080728564517468948234053907, −5.63618037162514355172420093297, −4.76356352391838759031409378696, −4.17775461403953723264187139486, −2.93543955322423669200344360772, −2.05465796941622513858995600804, −0.984132388000494402883300128060, 1.05174171415826649175345092694, 2.00302973821320356363173423201, 2.95568603735914240967997455321, 4.13777926104703638514442300344, 4.92299930210691171448404819404, 5.79764630779373397295473798533, 6.16480198992980972697105374008, 7.31494010911493843032727010991, 8.088542776011150673355778695998, 8.861596228876068996262075884445

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