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Properties

Label 2-2004-2004.2003-c0-0-15
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

\(L(\frac{1}{2})\) \(\approx\) \(1.354874869\)
\(L(\frac12)\) \(\approx\) \(1.354874869\)
\(L(1)\) 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

−9.153185246128506526850677113033, −8.879276186849607427647040622757, −7.987965655892556815219143801371, −7.07082369257477852149992716431, −6.46815055191435980337744499833, −5.59971734959201756791184949995, −4.36522854736731233247351960732, −3.02775246984645232156406584006, −2.25298910163709615521970108812, −1.52787103337297115316766718154, 1.52787103337297115316766718154, 2.25298910163709615521970108812, 3.02775246984645232156406584006, 4.36522854736731233247351960732, 5.59971734959201756791184949995, 6.46815055191435980337744499833, 7.07082369257477852149992716431, 7.987965655892556815219143801371, 8.879276186849607427647040622757, 9.153185246128506526850677113033

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