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Properties

Label 2-2005-2005.1603-c0-0-2
Degree $2$
Conductor $2005$
Sign $0.762 - 0.646i$
Analytic cond. $1.00062$
Root an. cond. $1.00031$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.642i)2-s + 0.175i·4-s + (0.309 − 0.951i)5-s + (0.221 − 0.221i)7-s + (−0.754 − 0.754i)8-s + i·9-s + (0.412 + 0.809i)10-s + 1.17·11-s + 0.284i·14-s + 0.793·16-s + (−0.642 − 0.642i)18-s + (0.166 + 0.0542i)20-s + (−0.754 + 0.754i)22-s + (−0.809 − 0.587i)25-s + (0.0388 + 0.0388i)28-s − 0.618i·29-s + ⋯
L(s)  = 1  + (−0.642 + 0.642i)2-s + 0.175i·4-s + (0.309 − 0.951i)5-s + (0.221 − 0.221i)7-s + (−0.754 − 0.754i)8-s + i·9-s + (0.412 + 0.809i)10-s + 1.17·11-s + 0.284i·14-s + 0.793·16-s + (−0.642 − 0.642i)18-s + (0.166 + 0.0542i)20-s + (−0.754 + 0.754i)22-s + (−0.809 − 0.587i)25-s + (0.0388 + 0.0388i)28-s − 0.618i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(1.00062\)
Root analytic conductor: \(1.00031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2005} (1603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2005,\ (\ :0),\ 0.762 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9278535213\)
\(L(\frac12)\) \(\approx\) \(0.9278535213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
401 \( 1 - T \)
good2 \( 1 + (0.642 - 0.642i)T - iT^{2} \)
3 \( 1 - iT^{2} \)
7 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
11 \( 1 - 1.17T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + 0.618iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.90T + T^{2} \)
43 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
47 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 + 1.17iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.261660715104661043737689441053, −8.603106916404153350538629286485, −7.926597270247881354149877091133, −7.32833720580682438835184694408, −6.35259651388829895980757276292, −5.62696099127081879084074768104, −4.54221079825556269151561969258, −3.88003993591884957925208687080, −2.40979735187258661593364543451, −1.10989846738800845030877744023, 1.13706067130765989634132740158, 2.21564714874350241352572035978, 3.17767250583146589038094486352, 4.07715377850195046417527964031, 5.46155929361590857585629843159, 6.22672096359795332466660994018, 6.76189443391133643513742970269, 7.79225339204170645201114739384, 8.964112890829799029540058598220, 9.245503761834245029268197495045

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