Properties

Label 2-8008-1.1-c1-0-5
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  − 3.34·3-s + 1.86·5-s − 7-s + 8.16·9-s − 11-s + 13-s − 6.22·15-s − 7.58·17-s − 7.29·19-s + 3.34·21-s − 5.61·23-s − 1.53·25-s − 17.2·27-s − 1.82·29-s + 2.16·31-s + 3.34·33-s − 1.86·35-s + 6.92·37-s − 3.34·39-s − 9.19·41-s − 4.98·43-s + 15.2·45-s + 6.13·47-s + 49-s + 25.3·51-s + 6.23·53-s − 1.86·55-s + ⋯
L(s)  = 1  − 1.92·3-s + 0.832·5-s − 0.377·7-s + 2.72·9-s − 0.301·11-s + 0.277·13-s − 1.60·15-s − 1.83·17-s − 1.67·19-s + 0.729·21-s − 1.17·23-s − 0.306·25-s − 3.32·27-s − 0.339·29-s + 0.388·31-s + 0.581·33-s − 0.314·35-s + 1.13·37-s − 0.535·39-s − 1.43·41-s − 0.760·43-s + 2.26·45-s + 0.894·47-s + 0.142·49-s + 3.54·51-s + 0.856·53-s − 0.251·55-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

\(L(1)\) \(\approx\) \(0.4275902884\)
\(L(\frac12)\) \(\approx\) \(0.4275902884\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.34T + 3T^{2} \)
5 \( 1 - 1.86T + 5T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + 7.29T + 19T^{2} \)
23 \( 1 + 5.61T + 23T^{2} \)
29 \( 1 + 1.82T + 29T^{2} \)
31 \( 1 - 2.16T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + 9.19T + 41T^{2} \)
43 \( 1 + 4.98T + 43T^{2} \)
47 \( 1 - 6.13T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 5.08T + 61T^{2} \)
67 \( 1 - 7.54T + 67T^{2} \)
71 \( 1 - 2.10T + 71T^{2} \)
73 \( 1 + 6.78T + 73T^{2} \)
79 \( 1 + 9.34T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 4.25T + 97T^{2} \)
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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

−7.60366359079489415668666352361, −6.71096638600642807384540636380, −6.24534713936896211583084816495, −6.04424385448585600652040197086, −5.12745103789879714251661881843, −4.46982065110518211795142447085, −3.90533902843702204189413726793, −2.29287130879354816587372888726, −1.72363964638603227026709092224, −0.34779841176542600102144785475, 0.34779841176542600102144785475, 1.72363964638603227026709092224, 2.29287130879354816587372888726, 3.90533902843702204189413726793, 4.46982065110518211795142447085, 5.12745103789879714251661881843, 6.04424385448585600652040197086, 6.24534713936896211583084816495, 6.71096638600642807384540636380, 7.60366359079489415668666352361

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