Properties

Label 2-4005-1.1-c1-0-35
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  − 1.30·2-s − 0.285·4-s + 5-s + 4.28·7-s + 2.99·8-s − 1.30·10-s − 3.71·11-s + 1.17·13-s − 5.60·14-s − 3.34·16-s − 7.72·17-s − 0.216·19-s − 0.285·20-s + 4.86·22-s + 1.27·23-s + 25-s − 1.54·26-s − 1.22·28-s + 5.32·29-s − 3.94·31-s − 1.60·32-s + 10.1·34-s + 4.28·35-s + 0.356·37-s + 0.283·38-s + 2.99·40-s + 11.1·41-s + ⋯
L(s)  = 1  − 0.925·2-s − 0.142·4-s + 0.447·5-s + 1.61·7-s + 1.05·8-s − 0.414·10-s − 1.11·11-s + 0.327·13-s − 1.49·14-s − 0.836·16-s − 1.87·17-s − 0.0496·19-s − 0.0639·20-s + 1.03·22-s + 0.266·23-s + 0.200·25-s − 0.302·26-s − 0.231·28-s + 0.989·29-s − 0.709·31-s − 0.283·32-s + 1.73·34-s + 0.723·35-s + 0.0586·37-s + 0.0459·38-s + 0.473·40-s + 1.73·41-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

\(L(1)\) \(\approx\) \(1.203598675\)
\(L(\frac12)\) \(\approx\) \(1.203598675\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.30T + 2T^{2} \)
7 \( 1 - 4.28T + 7T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 + 7.72T + 17T^{2} \)
19 \( 1 + 0.216T + 19T^{2} \)
23 \( 1 - 1.27T + 23T^{2} \)
29 \( 1 - 5.32T + 29T^{2} \)
31 \( 1 + 3.94T + 31T^{2} \)
37 \( 1 - 0.356T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 - 3.34T + 43T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + 5.58T + 53T^{2} \)
59 \( 1 - 1.34T + 59T^{2} \)
61 \( 1 - 3.59T + 61T^{2} \)
67 \( 1 - 8.60T + 67T^{2} \)
71 \( 1 - 7.59T + 71T^{2} \)
73 \( 1 - 3.56T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
97 \( 1 - 7.73T + 97T^{2} \)
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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

−8.478802637532919847775461558216, −7.936791561208694562089923811483, −7.30341240680289764955543545514, −6.36104404347815590242927348379, −5.26023700476312286134412066557, −4.79264986346817339430276485618, −4.09246151231686625211332843477, −2.47819702425603774511785989748, −1.86059069091535710308164456738, −0.74137341731732798148969113237, 0.74137341731732798148969113237, 1.86059069091535710308164456738, 2.47819702425603774511785989748, 4.09246151231686625211332843477, 4.79264986346817339430276485618, 5.26023700476312286134412066557, 6.36104404347815590242927348379, 7.30341240680289764955543545514, 7.936791561208694562089923811483, 8.478802637532919847775461558216

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