Properties

Label 2-8008-1.1-c1-0-37
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  + 1.69·3-s − 2.62·5-s − 7-s − 0.136·9-s − 11-s + 13-s − 4.44·15-s + 2.10·17-s + 4.75·19-s − 1.69·21-s + 7.30·23-s + 1.89·25-s − 5.30·27-s − 8.99·29-s + 4.35·31-s − 1.69·33-s + 2.62·35-s − 10.9·37-s + 1.69·39-s − 2.97·41-s − 6.62·43-s + 0.358·45-s + 8.67·47-s + 49-s + 3.56·51-s + 11.5·53-s + 2.62·55-s + ⋯
L(s)  = 1  + 0.976·3-s − 1.17·5-s − 0.377·7-s − 0.0455·9-s − 0.301·11-s + 0.277·13-s − 1.14·15-s + 0.510·17-s + 1.09·19-s − 0.369·21-s + 1.52·23-s + 0.379·25-s − 1.02·27-s − 1.67·29-s + 0.782·31-s − 0.294·33-s + 0.443·35-s − 1.80·37-s + 0.270·39-s − 0.463·41-s − 1.00·43-s + 0.0534·45-s + 1.26·47-s + 0.142·49-s + 0.498·51-s + 1.58·53-s + 0.354·55-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

\(L(1)\) \(\approx\) \(1.807956538\)
\(L(\frac12)\) \(\approx\) \(1.807956538\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.69T + 3T^{2} \)
5 \( 1 + 2.62T + 5T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 - 7.30T + 23T^{2} \)
29 \( 1 + 8.99T + 29T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 2.97T + 41T^{2} \)
43 \( 1 + 6.62T + 43T^{2} \)
47 \( 1 - 8.67T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 8.45T + 59T^{2} \)
61 \( 1 + 2.44T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 - 8.55T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 - 0.0240T + 89T^{2} \)
97 \( 1 - 3.50T + 97T^{2} \)
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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

−7.73666319354062302839548798702, −7.43889409787218971895220324252, −6.71041999411053962451959723315, −5.60120320408238291566343777085, −5.05651920725345903224007481845, −3.96291795831526449273244432987, −3.36745940633454713227639189035, −3.03430339127301798277277114861, −1.88713200104176224223087582766, −0.62100668057755317296916496609, 0.62100668057755317296916496609, 1.88713200104176224223087582766, 3.03430339127301798277277114861, 3.36745940633454713227639189035, 3.96291795831526449273244432987, 5.05651920725345903224007481845, 5.60120320408238291566343777085, 6.71041999411053962451959723315, 7.43889409787218971895220324252, 7.73666319354062302839548798702

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