Properties

Label 2-8008-1.1-c1-0-22
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  − 1.59·3-s + 0.804·5-s − 7-s − 0.458·9-s + 11-s + 13-s − 1.28·15-s − 0.398·17-s − 1.98·19-s + 1.59·21-s − 3.65·23-s − 4.35·25-s + 5.51·27-s − 0.285·29-s + 3.90·31-s − 1.59·33-s − 0.804·35-s − 11.0·37-s − 1.59·39-s + 4.88·41-s + 1.43·43-s − 0.368·45-s + 0.275·47-s + 49-s + 0.634·51-s − 3.81·53-s + 0.804·55-s + ⋯
L(s)  = 1  − 0.920·3-s + 0.359·5-s − 0.377·7-s − 0.152·9-s + 0.301·11-s + 0.277·13-s − 0.331·15-s − 0.0965·17-s − 0.455·19-s + 0.347·21-s − 0.761·23-s − 0.870·25-s + 1.06·27-s − 0.0530·29-s + 0.701·31-s − 0.277·33-s − 0.135·35-s − 1.81·37-s − 0.255·39-s + 0.762·41-s + 0.219·43-s − 0.0549·45-s + 0.0402·47-s + 0.142·49-s + 0.0888·51-s − 0.523·53-s + 0.108·55-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

\(L(1)\) \(\approx\) \(0.9788859308\)
\(L(\frac12)\) \(\approx\) \(0.9788859308\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.59T + 3T^{2} \)
5 \( 1 - 0.804T + 5T^{2} \)
17 \( 1 + 0.398T + 17T^{2} \)
19 \( 1 + 1.98T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 0.285T + 29T^{2} \)
31 \( 1 - 3.90T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 - 1.43T + 43T^{2} \)
47 \( 1 - 0.275T + 47T^{2} \)
53 \( 1 + 3.81T + 53T^{2} \)
59 \( 1 - 8.19T + 59T^{2} \)
61 \( 1 + 7.08T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 2.28T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 3.20T + 79T^{2} \)
83 \( 1 - 4.69T + 83T^{2} \)
89 \( 1 + 2.84T + 89T^{2} \)
97 \( 1 - 2.50T + 97T^{2} \)
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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

−7.88225411719757555191658472876, −6.78386798992178839916544971276, −6.47952494146792575211324782826, −5.72698673278601147098927069274, −5.28567013843178683344509500589, −4.31985673511933413164120291146, −3.62422894824313690197132829375, −2.61098232213773353702685824437, −1.69453497619281752773748076084, −0.50868390065528439472583494855, 0.50868390065528439472583494855, 1.69453497619281752773748076084, 2.61098232213773353702685824437, 3.62422894824313690197132829375, 4.31985673511933413164120291146, 5.28567013843178683344509500589, 5.72698673278601147098927069274, 6.47952494146792575211324782826, 6.78386798992178839916544971276, 7.88225411719757555191658472876

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