Properties

Label 2-8008-1.1-c1-0-55
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  + 1.61·3-s − 0.618·5-s − 7-s − 0.381·9-s + 11-s − 13-s − 1.00·15-s + 0.854·17-s + 7.09·19-s − 1.61·21-s + 2.47·23-s − 4.61·25-s − 5.47·27-s + 1.61·33-s + 0.618·35-s + 2.76·37-s − 1.61·39-s − 11.7·41-s + 4.09·43-s + 0.236·45-s − 4·47-s + 49-s + 1.38·51-s + 7.61·53-s − 0.618·55-s + 11.4·57-s + 6·59-s + ⋯
L(s)  = 1  + 0.934·3-s − 0.276·5-s − 0.377·7-s − 0.127·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.207·17-s + 1.62·19-s − 0.353·21-s + 0.515·23-s − 0.923·25-s − 1.05·27-s + 0.281·33-s + 0.104·35-s + 0.454·37-s − 0.259·39-s − 1.82·41-s + 0.623·43-s + 0.0351·45-s − 0.583·47-s + 0.142·49-s + 0.193·51-s + 1.04·53-s − 0.0833·55-s + 1.51·57-s + 0.781·59-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

\(L(1)\) \(\approx\) \(2.449061953\)
\(L(\frac12)\) \(\approx\) \(2.449061953\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 3T^{2} \)
5 \( 1 + 0.618T + 5T^{2} \)
17 \( 1 - 0.854T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2.76T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 4.09T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 5.70T + 73T^{2} \)
79 \( 1 - 4.61T + 79T^{2} \)
83 \( 1 - 6.14T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.966619582513237075922645392857, −7.21519805821616025312639012810, −6.66528475089868918943734070071, −5.60999900109687516492151414278, −5.15151557811363224353330832781, −3.97206369200633276632331992072, −3.47333591434763346922513216324, −2.79781088439245053555135025834, −1.94136736697565553688181183716, −0.73081687646427890641279274082, 0.73081687646427890641279274082, 1.94136736697565553688181183716, 2.79781088439245053555135025834, 3.47333591434763346922513216324, 3.97206369200633276632331992072, 5.15151557811363224353330832781, 5.60999900109687516492151414278, 6.66528475089868918943734070071, 7.21519805821616025312639012810, 7.966619582513237075922645392857

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