Properties

Label 2-2004-2004.2003-c0-0-22
Degree $2$
Conductor $2004$
Sign $1$
Analytic cond. $1.00012$
Root an. cond. $1.00006$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s + 16-s + 18-s − 2·22-s + 24-s − 25-s + 27-s + 32-s − 2·33-s + 36-s − 2·44-s − 2·47-s + 48-s + 49-s − 50-s + 54-s − 2·61-s + 64-s − 2·66-s + 72-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s + 16-s + 18-s − 2·22-s + 24-s − 25-s + 27-s + 32-s − 2·33-s + 36-s − 2·44-s − 2·47-s + 48-s + 49-s − 50-s + 54-s − 2·61-s + 64-s − 2·66-s + 72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(1.00012\)
Root analytic conductor: \(1.00006\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2004} (2003, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2004,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.828944251\)
\(L(\frac12)\) \(\approx\) \(2.828944251\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 + T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( ( 1 + T )^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
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

−9.409026819030253402152017068599, −8.210781164323262641046444724904, −7.81344714850669165146848639095, −7.10326380417237278822140142446, −6.05523110315333649129783722238, −5.17015322538927603798048442380, −4.44788671815030718172705710070, −3.40351519948223835697325871832, −2.70002137765165628169346249819, −1.85390657607385376562342581241, 1.85390657607385376562342581241, 2.70002137765165628169346249819, 3.40351519948223835697325871832, 4.44788671815030718172705710070, 5.17015322538927603798048442380, 6.05523110315333649129783722238, 7.10326380417237278822140142446, 7.81344714850669165146848639095, 8.210781164323262641046444724904, 9.409026819030253402152017068599

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