Properties

Label 2-2005-2005.1603-c0-0-0
Degree $2$
Conductor $2005$
Sign $-0.997 + 0.0746i$
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  + (−0.221 + 0.221i)2-s + 0.902i·4-s + (−0.809 + 0.587i)5-s + (−1.26 + 1.26i)7-s + (−0.420 − 0.420i)8-s + i·9-s + (0.0489 − 0.309i)10-s + 1.90·11-s − 0.557i·14-s − 0.715·16-s + (−0.221 − 0.221i)18-s + (−0.530 − 0.729i)20-s + (−0.420 + 0.420i)22-s + (0.309 − 0.951i)25-s + (−1.13 − 1.13i)28-s + 1.61i·29-s + ⋯
L(s)  = 1  + (−0.221 + 0.221i)2-s + 0.902i·4-s + (−0.809 + 0.587i)5-s + (−1.26 + 1.26i)7-s + (−0.420 − 0.420i)8-s + i·9-s + (0.0489 − 0.309i)10-s + 1.90·11-s − 0.557i·14-s − 0.715·16-s + (−0.221 − 0.221i)18-s + (−0.530 − 0.729i)20-s + (−0.420 + 0.420i)22-s + (0.309 − 0.951i)25-s + (−1.13 − 1.13i)28-s + 1.61i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $-0.997 + 0.0746i$
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: \((2,\ 2005,\ (\ :0),\ -0.997 + 0.0746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6436442907\)
\(L(\frac12)\) \(\approx\) \(0.6436442907\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
401 \( 1 - T \)
good2 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
3 \( 1 - iT^{2} \)
7 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
11 \( 1 - 1.90T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.61iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 1.17T + T^{2} \)
43 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
47 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 + 1.90iT - T^{2} \)
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

−9.501812733108901561795262767516, −8.736035482258990262967791009762, −8.435248736664700291600643860658, −7.19289616368012617379592877014, −6.83166704674457527355506779711, −6.08104417921846240972233850667, −4.84553566646704005869962753647, −3.62026032227337253544220732873, −3.29839925249418440758827739939, −2.15253366149485497566265419712, 0.53657553587055374050787212647, 1.38964612056731521837406879468, 3.30393388211978263788813316756, 3.93110282302529875702259876435, 4.62518422797022208444440264964, 6.09404552042216246274973685327, 6.52683271736686964326739643977, 7.17440259158546449949241436748, 8.364663925314052524823332067168, 9.270202741791219067418993373840

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