Properties

Label 2-4005-1.1-c1-0-99
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.875·2-s − 1.23·4-s − 5-s − 3.38·7-s − 2.83·8-s − 0.875·10-s + 2.67·11-s + 2.37·13-s − 2.96·14-s − 0.0143·16-s + 5.74·17-s − 0.615·19-s + 1.23·20-s + 2.34·22-s + 3.22·23-s + 25-s + 2.08·26-s + 4.17·28-s − 6.94·29-s − 2.63·31-s + 5.65·32-s + 5.02·34-s + 3.38·35-s + 6.16·37-s − 0.538·38-s + 2.83·40-s + 8.24·41-s + ⋯
L(s)  = 1  + 0.619·2-s − 0.616·4-s − 0.447·5-s − 1.27·7-s − 1.00·8-s − 0.276·10-s + 0.805·11-s + 0.659·13-s − 0.792·14-s − 0.00357·16-s + 1.39·17-s − 0.141·19-s + 0.275·20-s + 0.498·22-s + 0.672·23-s + 0.200·25-s + 0.408·26-s + 0.788·28-s − 1.28·29-s − 0.474·31-s + 0.998·32-s + 0.862·34-s + 0.572·35-s + 1.01·37-s − 0.0873·38-s + 0.447·40-s + 1.28·41-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: \(1\)
Selberg data: \((2,\ 4005,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.875T + 2T^{2} \)
7 \( 1 + 3.38T + 7T^{2} \)
11 \( 1 - 2.67T + 11T^{2} \)
13 \( 1 - 2.37T + 13T^{2} \)
17 \( 1 - 5.74T + 17T^{2} \)
19 \( 1 + 0.615T + 19T^{2} \)
23 \( 1 - 3.22T + 23T^{2} \)
29 \( 1 + 6.94T + 29T^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 - 6.16T + 37T^{2} \)
41 \( 1 - 8.24T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 8.61T + 61T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 + 9.34T + 71T^{2} \)
73 \( 1 + 2.05T + 73T^{2} \)
79 \( 1 - 8.28T + 79T^{2} \)
83 \( 1 + 4.56T + 83T^{2} \)
97 \( 1 + 1.11T + 97T^{2} \)
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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

−8.073663346209452568921095357932, −7.30707100788284909944536326915, −6.31270576958090706236145831290, −5.96206026699312491766374481759, −5.02556237306916964816636877857, −4.10584111618756257932083417106, −3.46296674394052872141356044820, −3.03260045694811092101155656700, −1.27152147015427614634012060463, 0, 1.27152147015427614634012060463, 3.03260045694811092101155656700, 3.46296674394052872141356044820, 4.10584111618756257932083417106, 5.02556237306916964816636877857, 5.96206026699312491766374481759, 6.31270576958090706236145831290, 7.30707100788284909944536326915, 8.073663346209452568921095357932

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