Properties

Label 2-2008-2008.501-c0-0-1
Degree $2$
Conductor $2008$
Sign $1$
Analytic cond. $1.00212$
Root an. cond. $1.00106$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 0.445·7-s − 8-s + 9-s + 1.80·11-s + 0.445·14-s + 16-s + 1.24·17-s − 18-s − 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s − 1.24·29-s + 1.24·31-s − 32-s − 1.24·34-s + 36-s + 0.445·37-s + 1.24·38-s − 1.80·41-s + 0.445·43-s + 1.80·44-s + 1.80·46-s − 0.801·49-s + ⋯
L(s)  = 1  − 2-s + 4-s − 0.445·7-s − 8-s + 9-s + 1.80·11-s + 0.445·14-s + 16-s + 1.24·17-s − 18-s − 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s − 1.24·29-s + 1.24·31-s − 32-s − 1.24·34-s + 36-s + 0.445·37-s + 1.24·38-s − 1.80·41-s + 0.445·43-s + 1.80·44-s + 1.80·46-s − 0.801·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $1$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (501, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8571470285\)
\(L(\frac12)\) \(\approx\) \(0.8571470285\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
251 \( 1 + T \)
good3 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 0.445T + T^{2} \)
11 \( 1 - 1.80T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 + 1.24T + T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 + 1.24T + T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 - 0.445T + T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
43 \( 1 - 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.80T + T^{2} \)
59 \( 1 - 0.445T + T^{2} \)
61 \( 1 - 1.80T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.445T + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 0.445T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.471545750207664100747207159575, −8.587089118816356606508794573300, −7.943906029771986606418082809963, −6.85032389312096640173458943905, −6.60059441440115932412898267055, −5.66078732644009157287456966480, −4.17007871439038847074469495222, −3.54462451090239104262059073847, −2.11344967430654172826563837789, −1.15651804087806272246957086694, 1.15651804087806272246957086694, 2.11344967430654172826563837789, 3.54462451090239104262059073847, 4.17007871439038847074469495222, 5.66078732644009157287456966480, 6.60059441440115932412898267055, 6.85032389312096640173458943905, 7.943906029771986606418082809963, 8.587089118816356606508794573300, 9.471545750207664100747207159575

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