Properties

Label 2-8008-1.1-c1-0-168
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.79·3-s + 0.876·5-s + 7-s + 0.233·9-s − 11-s + 13-s + 1.57·15-s − 1.98·17-s − 7.48·19-s + 1.79·21-s − 5.93·23-s − 4.23·25-s − 4.97·27-s + 4.29·29-s + 3.81·31-s − 1.79·33-s + 0.876·35-s − 4.36·37-s + 1.79·39-s − 4.55·41-s + 12.2·43-s + 0.204·45-s + 8.41·47-s + 49-s − 3.56·51-s + 1.66·53-s − 0.876·55-s + ⋯
L(s)  = 1  + 1.03·3-s + 0.391·5-s + 0.377·7-s + 0.0776·9-s − 0.301·11-s + 0.277·13-s + 0.406·15-s − 0.480·17-s − 1.71·19-s + 0.392·21-s − 1.23·23-s − 0.846·25-s − 0.957·27-s + 0.797·29-s + 0.685·31-s − 0.313·33-s + 0.148·35-s − 0.718·37-s + 0.287·39-s − 0.711·41-s + 1.86·43-s + 0.0304·45-s + 1.22·47-s + 0.142·49-s − 0.499·51-s + 0.228·53-s − 0.118·55-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: \(1\)
Selberg data: \((2,\ 8008,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.79T + 3T^{2} \)
5 \( 1 - 0.876T + 5T^{2} \)
17 \( 1 + 1.98T + 17T^{2} \)
19 \( 1 + 7.48T + 19T^{2} \)
23 \( 1 + 5.93T + 23T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 - 3.81T + 31T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 - 8.41T + 47T^{2} \)
53 \( 1 - 1.66T + 53T^{2} \)
59 \( 1 + 8.44T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 - 6.77T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 4.82T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−7.77238758342398580758827097504, −6.82217339102436900750669001757, −6.07281700351120935357504264823, −5.53875556797544430504614145762, −4.31026510035181651585103603380, −4.07126257939552852277801591111, −2.87287712171161656681100311730, −2.31494222719130867085654478908, −1.61014566850716425107217072364, 0, 1.61014566850716425107217072364, 2.31494222719130867085654478908, 2.87287712171161656681100311730, 4.07126257939552852277801591111, 4.31026510035181651585103603380, 5.53875556797544430504614145762, 6.07281700351120935357504264823, 6.82217339102436900750669001757, 7.77238758342398580758827097504

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