Properties

Label 2-4005-1.1-c1-0-47
Degree $2$
Conductor $4005$
Sign $1$
Analytic cond. $31.9800$
Root an. cond. $5.65509$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.395·2-s − 1.84·4-s + 5-s + 1.55·7-s − 1.52·8-s + 0.395·10-s + 3.04·11-s + 1.37·13-s + 0.616·14-s + 3.08·16-s − 0.217·17-s − 0.596·19-s − 1.84·20-s + 1.20·22-s + 2.26·23-s + 25-s + 0.544·26-s − 2.87·28-s − 3.13·29-s + 8.00·31-s + 4.26·32-s − 0.0861·34-s + 1.55·35-s + 7.10·37-s − 0.236·38-s − 1.52·40-s − 8.90·41-s + ⋯
L(s)  = 1  + 0.279·2-s − 0.921·4-s + 0.447·5-s + 0.588·7-s − 0.537·8-s + 0.125·10-s + 0.918·11-s + 0.381·13-s + 0.164·14-s + 0.771·16-s − 0.0528·17-s − 0.136·19-s − 0.412·20-s + 0.257·22-s + 0.472·23-s + 0.200·25-s + 0.106·26-s − 0.542·28-s − 0.582·29-s + 1.43·31-s + 0.753·32-s − 0.0147·34-s + 0.263·35-s + 1.16·37-s − 0.0383·38-s − 0.240·40-s − 1.39·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
Sign: $1$
Analytic conductor: \(31.9800\)
Root analytic conductor: \(5.65509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4005,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.188279560\)
\(L(\frac12)\) \(\approx\) \(2.188279560\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
89 \( 1 - T \)
good2 \( 1 - 0.395T + 2T^{2} \)
7 \( 1 - 1.55T + 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 + 0.217T + 17T^{2} \)
19 \( 1 + 0.596T + 19T^{2} \)
23 \( 1 - 2.26T + 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 - 8.00T + 31T^{2} \)
37 \( 1 - 7.10T + 37T^{2} \)
41 \( 1 + 8.90T + 41T^{2} \)
43 \( 1 + 9.92T + 43T^{2} \)
47 \( 1 + 4.37T + 47T^{2} \)
53 \( 1 + 0.711T + 53T^{2} \)
59 \( 1 + 0.315T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 - 1.38T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 7.82T + 83T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−8.426090377571242868362136044344, −7.996711600471897511697943653246, −6.74001045731149234070938014955, −6.26204294631762185080286143930, −5.26083670159467417474640383998, −4.77081495157775948731686084627, −3.92169183110405575218481933130, −3.16286908168050699043863523646, −1.86971267501799714660846750921, −0.863198019118501621926982470855, 0.863198019118501621926982470855, 1.86971267501799714660846750921, 3.16286908168050699043863523646, 3.92169183110405575218481933130, 4.77081495157775948731686084627, 5.26083670159467417474640383998, 6.26204294631762185080286143930, 6.74001045731149234070938014955, 7.996711600471897511697943653246, 8.426090377571242868362136044344

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