Properties

Label 2-6024-1.1-c1-0-40
Degree $2$
Conductor $6024$
Sign $1$
Analytic cond. $48.1018$
Root an. cond. $6.93555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.55·5-s + 2.50·7-s + 9-s − 4.27·11-s − 0.800·13-s − 3.55·15-s + 4.44·17-s + 3.36·19-s − 2.50·21-s − 6.70·23-s + 7.65·25-s − 27-s + 5.98·29-s − 0.492·31-s + 4.27·33-s + 8.91·35-s − 4.22·37-s + 0.800·39-s − 2.63·41-s − 5.98·43-s + 3.55·45-s + 10.5·47-s − 0.717·49-s − 4.44·51-s + 4.59·53-s − 15.1·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.59·5-s + 0.947·7-s + 0.333·9-s − 1.28·11-s − 0.222·13-s − 0.918·15-s + 1.07·17-s + 0.772·19-s − 0.546·21-s − 1.39·23-s + 1.53·25-s − 0.192·27-s + 1.11·29-s − 0.0884·31-s + 0.743·33-s + 1.50·35-s − 0.694·37-s + 0.128·39-s − 0.412·41-s − 0.912·43-s + 0.530·45-s + 1.54·47-s − 0.102·49-s − 0.622·51-s + 0.631·53-s − 2.04·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6024\)    =    \(2^{3} \cdot 3 \cdot 251\)
Sign: $1$
Analytic conductor: \(48.1018\)
Root analytic conductor: \(6.93555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.500047833\)
\(L(\frac12)\) \(\approx\) \(2.500047833\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
251 \( 1 - T \)
good5 \( 1 - 3.55T + 5T^{2} \)
7 \( 1 - 2.50T + 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + 0.800T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 - 3.36T + 19T^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 - 5.98T + 29T^{2} \)
31 \( 1 + 0.492T + 31T^{2} \)
37 \( 1 + 4.22T + 37T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 + 5.98T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 4.59T + 53T^{2} \)
59 \( 1 + 4.66T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 1.63T + 67T^{2} \)
71 \( 1 - 7.10T + 71T^{2} \)
73 \( 1 - 9.50T + 73T^{2} \)
79 \( 1 - 6.01T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−8.012868944318591029425674508892, −7.41989183792417114439208637041, −6.47545224194861308443420431484, −5.80216157762288156532616441807, −5.13084710258833016680167941890, −5.00079736942272903493953538788, −3.62264997967696130552314277000, −2.48599243054602807137966440612, −1.88771412296436141648812937283, −0.882444991530166200821786094582, 0.882444991530166200821786094582, 1.88771412296436141648812937283, 2.48599243054602807137966440612, 3.62264997967696130552314277000, 5.00079736942272903493953538788, 5.13084710258833016680167941890, 5.80216157762288156532616441807, 6.47545224194861308443420431484, 7.41989183792417114439208637041, 8.012868944318591029425674508892

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