Properties

Label 2-2004-2004.2003-c0-0-9
Degree $2$
Conductor $2004$
Sign $1$
Analytic cond. $1.00012$
Root an. cond. $1.00006$
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 − 3-s + 4-s − 1.41·5-s − 6-s + 8-s + 9-s − 1.41·10-s − 12-s + 1.41·15-s + 16-s + 1.41·17-s + 18-s − 1.41·20-s − 24-s + 1.00·25-s − 27-s + 1.41·30-s + 32-s + 1.41·34-s + 36-s − 1.41·40-s + 1.41·41-s + 1.41·43-s − 1.41·45-s − 48-s + 49-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 1.41·5-s − 6-s + 8-s + 9-s − 1.41·10-s − 12-s + 1.41·15-s + 16-s + 1.41·17-s + 18-s − 1.41·20-s − 24-s + 1.00·25-s − 27-s + 1.41·30-s + 32-s + 1.41·34-s + 36-s − 1.41·40-s + 1.41·41-s + 1.41·43-s − 1.41·45-s − 48-s + 49-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: no
Arithmetic: yes
Character: $\chi_{2004} (2003, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2004,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.366796737\)
\(L(\frac12)\) \(\approx\) \(1.366796737\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.536829639078312526742371409379, −8.184645454819341045405136736520, −7.46807052328453119794748255448, −7.04374713855224369388064666607, −5.90570503240493090306770099654, −5.40200547686342284004699658787, −4.28647837715319173215694718811, −3.94448536080837266151985552466, −2.80310220339979786929606683550, −1.11827589940167191063800938637, 1.11827589940167191063800938637, 2.80310220339979786929606683550, 3.94448536080837266151985552466, 4.28647837715319173215694718811, 5.40200547686342284004699658787, 5.90570503240493090306770099654, 7.04374713855224369388064666607, 7.46807052328453119794748255448, 8.184645454819341045405136736520, 9.536829639078312526742371409379

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