Properties

Label 2-6008-1.1-c1-0-121
Degree $2$
Conductor $6008$
Sign $1$
Analytic cond. $47.9741$
Root an. cond. $6.92633$
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  + 2.51·3-s + 0.436·5-s + 4.63·7-s + 3.31·9-s + 0.311·11-s + 4.20·13-s + 1.09·15-s − 6.15·17-s + 2.13·19-s + 11.6·21-s + 2.66·23-s − 4.80·25-s + 0.796·27-s + 4.83·29-s + 8.22·31-s + 0.782·33-s + 2.02·35-s − 0.726·37-s + 10.5·39-s + 7.75·41-s − 9.66·43-s + 1.44·45-s − 9.99·47-s + 14.4·49-s − 15.4·51-s + 6.02·53-s + 0.135·55-s + ⋯
L(s)  = 1  + 1.45·3-s + 0.195·5-s + 1.75·7-s + 1.10·9-s + 0.0938·11-s + 1.16·13-s + 0.283·15-s − 1.49·17-s + 0.489·19-s + 2.54·21-s + 0.554·23-s − 0.961·25-s + 0.153·27-s + 0.897·29-s + 1.47·31-s + 0.136·33-s + 0.342·35-s − 0.119·37-s + 1.69·39-s + 1.21·41-s − 1.47·43-s + 0.215·45-s − 1.45·47-s + 2.06·49-s − 2.16·51-s + 0.826·53-s + 0.0183·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $1$
Analytic conductor: \(47.9741\)
Root analytic conductor: \(6.92633\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.837268220\)
\(L(\frac12)\) \(\approx\) \(4.837268220\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 - T \)
good3 \( 1 - 2.51T + 3T^{2} \)
5 \( 1 - 0.436T + 5T^{2} \)
7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 - 0.311T + 11T^{2} \)
13 \( 1 - 4.20T + 13T^{2} \)
17 \( 1 + 6.15T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 - 2.66T + 23T^{2} \)
29 \( 1 - 4.83T + 29T^{2} \)
31 \( 1 - 8.22T + 31T^{2} \)
37 \( 1 + 0.726T + 37T^{2} \)
41 \( 1 - 7.75T + 41T^{2} \)
43 \( 1 + 9.66T + 43T^{2} \)
47 \( 1 + 9.99T + 47T^{2} \)
53 \( 1 - 6.02T + 53T^{2} \)
59 \( 1 + 9.95T + 59T^{2} \)
61 \( 1 + 6.81T + 61T^{2} \)
67 \( 1 - 2.42T + 67T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 + 6.72T + 73T^{2} \)
79 \( 1 - 6.86T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 + 3.08T + 89T^{2} \)
97 \( 1 - 3.63T + 97T^{2} \)
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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

−8.224535130556721120231959419983, −7.71209158772440852140239065078, −6.79369352921978400277333728582, −6.02287063978313808857928506917, −4.91385900131278144101518342027, −4.42774425335602308199608381919, −3.59649923803568722373703602661, −2.67572583904839449677978603309, −1.92252681456202468220224875247, −1.21881499393581153358461700262, 1.21881499393581153358461700262, 1.92252681456202468220224875247, 2.67572583904839449677978603309, 3.59649923803568722373703602661, 4.42774425335602308199608381919, 4.91385900131278144101518342027, 6.02287063978313808857928506917, 6.79369352921978400277333728582, 7.71209158772440852140239065078, 8.224535130556721120231959419983

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