Properties

Label 2-1008-1.1-c1-0-4
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 4·11-s + 6·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s + 2·29-s + 2·35-s − 10·37-s + 6·41-s + 4·43-s + 49-s − 6·53-s − 8·55-s + 4·59-s + 6·61-s + 12·65-s − 4·67-s + 8·71-s + 10·73-s − 4·77-s − 4·83-s − 4·85-s + 6·89-s + 6·91-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.338·35-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s − 0.439·83-s − 0.433·85-s + 0.635·89-s + 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.006550525\)
\(L(\frac12)\) \(\approx\) \(2.006550525\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.998713269577231931539325852863, −9.085422624242741773351689713512, −8.427817591055194605299416158333, −7.46998199559376075566652373057, −6.48878700424638503015501865218, −5.58110052520146908580921209772, −4.96240850233932644050318806427, −3.57539886636491016875562737772, −2.47676327970158091646809943718, −1.22168166913217334012046287130, 1.22168166913217334012046287130, 2.47676327970158091646809943718, 3.57539886636491016875562737772, 4.96240850233932644050318806427, 5.58110052520146908580921209772, 6.48878700424638503015501865218, 7.46998199559376075566652373057, 8.427817591055194605299416158333, 9.085422624242741773351689713512, 9.998713269577231931539325852863

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