Properties

Label 2-8008-1.1-c1-0-13
Degree $2$
Conductor $8008$
Sign $1$
Analytic cond. $63.9442$
Root an. cond. $7.99651$
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  − 1.74·3-s − 0.876·5-s − 7-s + 0.0606·9-s − 11-s + 13-s + 1.53·15-s + 2.59·17-s + 4.40·19-s + 1.74·21-s − 7.88·23-s − 4.23·25-s + 5.14·27-s − 5.50·29-s − 0.941·31-s + 1.74·33-s + 0.876·35-s − 1.26·37-s − 1.74·39-s + 10.7·41-s − 5.13·43-s − 0.0531·45-s − 5.09·47-s + 49-s − 4.53·51-s + 13.0·53-s + 0.876·55-s + ⋯
L(s)  = 1  − 1.01·3-s − 0.391·5-s − 0.377·7-s + 0.0202·9-s − 0.301·11-s + 0.277·13-s + 0.395·15-s + 0.628·17-s + 1.01·19-s + 0.381·21-s − 1.64·23-s − 0.846·25-s + 0.989·27-s − 1.02·29-s − 0.169·31-s + 0.304·33-s + 0.148·35-s − 0.207·37-s − 0.280·39-s + 1.68·41-s − 0.782·43-s − 0.00792·45-s − 0.743·47-s + 0.142·49-s − 0.634·51-s + 1.79·53-s + 0.118·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(63.9442\)
Root analytic conductor: \(7.99651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6258488542\)
\(L(\frac12)\) \(\approx\) \(0.6258488542\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 + 1.74T + 3T^{2} \)
5 \( 1 + 0.876T + 5T^{2} \)
17 \( 1 - 2.59T + 17T^{2} \)
19 \( 1 - 4.40T + 19T^{2} \)
23 \( 1 + 7.88T + 23T^{2} \)
29 \( 1 + 5.50T + 29T^{2} \)
31 \( 1 + 0.941T + 31T^{2} \)
37 \( 1 + 1.26T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 5.13T + 43T^{2} \)
47 \( 1 + 5.09T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + 1.26T + 59T^{2} \)
61 \( 1 + 9.35T + 61T^{2} \)
67 \( 1 - 1.11T + 67T^{2} \)
71 \( 1 + 8.01T + 71T^{2} \)
73 \( 1 + 8.99T + 73T^{2} \)
79 \( 1 - 6.51T + 79T^{2} \)
83 \( 1 + 9.40T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 2.80T + 97T^{2} \)
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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

−7.68137243762764739458851051593, −7.22785459901872593065794813636, −6.18230903527731610314971452220, −5.80561929572238266730146353666, −5.26958760200074961648300699520, −4.27703638717350634732505092063, −3.62256815268852992604483002874, −2.75875221727411586163429616837, −1.58930588195859875367805361133, −0.41502278432478924172171915260, 0.41502278432478924172171915260, 1.58930588195859875367805361133, 2.75875221727411586163429616837, 3.62256815268852992604483002874, 4.27703638717350634732505092063, 5.26958760200074961648300699520, 5.80561929572238266730146353666, 6.18230903527731610314971452220, 7.22785459901872593065794813636, 7.68137243762764739458851051593

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