Properties

Label 2-8008-1.1-c1-0-102
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  + 2.48·3-s + 1.56·5-s − 7-s + 3.15·9-s + 11-s + 13-s + 3.87·15-s + 1.17·17-s + 6.31·19-s − 2.48·21-s + 4.27·23-s − 2.56·25-s + 0.390·27-s − 6.71·29-s − 3.78·31-s + 2.48·33-s − 1.56·35-s + 2.14·37-s + 2.48·39-s + 11.7·41-s − 4.31·43-s + 4.92·45-s − 3.97·47-s + 49-s + 2.92·51-s + 4.15·53-s + 1.56·55-s + ⋯
L(s)  = 1  + 1.43·3-s + 0.698·5-s − 0.377·7-s + 1.05·9-s + 0.301·11-s + 0.277·13-s + 1.00·15-s + 0.285·17-s + 1.44·19-s − 0.541·21-s + 0.890·23-s − 0.512·25-s + 0.0752·27-s − 1.24·29-s − 0.680·31-s + 0.431·33-s − 0.263·35-s + 0.352·37-s + 0.397·39-s + 1.83·41-s − 0.657·43-s + 0.734·45-s − 0.579·47-s + 0.142·49-s + 0.409·51-s + 0.570·53-s + 0.210·55-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

\(L(1)\) \(\approx\) \(4.366352526\)
\(L(\frac12)\) \(\approx\) \(4.366352526\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.48T + 3T^{2} \)
5 \( 1 - 1.56T + 5T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 - 6.31T + 19T^{2} \)
23 \( 1 - 4.27T + 23T^{2} \)
29 \( 1 + 6.71T + 29T^{2} \)
31 \( 1 + 3.78T + 31T^{2} \)
37 \( 1 - 2.14T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 4.31T + 43T^{2} \)
47 \( 1 + 3.97T + 47T^{2} \)
53 \( 1 - 4.15T + 53T^{2} \)
59 \( 1 - 5.96T + 59T^{2} \)
61 \( 1 - 6.28T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + 0.0156T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 + 2.88T + 79T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 - 5.87T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−7.76528006942770604144727041250, −7.38151697167181273736637018800, −6.55805145303561023215797792282, −5.69475746344510435471859661051, −5.14826439902232917326422020201, −3.86574229555183158164542493490, −3.52345351135343307666220430315, −2.65598711605861066583276747183, −1.98642950358766981309839414651, −1.00685727346929222851253481678, 1.00685727346929222851253481678, 1.98642950358766981309839414651, 2.65598711605861066583276747183, 3.52345351135343307666220430315, 3.86574229555183158164542493490, 5.14826439902232917326422020201, 5.69475746344510435471859661051, 6.55805145303561023215797792282, 7.38151697167181273736637018800, 7.76528006942770604144727041250

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