Properties

Label 2-6002-1.1-c1-0-222
Degree $2$
Conductor $6002$
Sign $-1$
Analytic cond. $47.9262$
Root an. cond. $6.92287$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.0548·3-s + 4-s + 0.621·5-s − 0.0548·6-s + 1.57·7-s + 8-s − 2.99·9-s + 0.621·10-s − 4.75·11-s − 0.0548·12-s + 4.63·13-s + 1.57·14-s − 0.0340·15-s + 16-s − 4.97·17-s − 2.99·18-s − 2.09·19-s + 0.621·20-s − 0.0861·21-s − 4.75·22-s − 1.82·23-s − 0.0548·24-s − 4.61·25-s + 4.63·26-s + 0.328·27-s + 1.57·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.0316·3-s + 0.5·4-s + 0.277·5-s − 0.0223·6-s + 0.593·7-s + 0.353·8-s − 0.998·9-s + 0.196·10-s − 1.43·11-s − 0.0158·12-s + 1.28·13-s + 0.419·14-s − 0.00879·15-s + 0.250·16-s − 1.20·17-s − 0.706·18-s − 0.480·19-s + 0.138·20-s − 0.0187·21-s − 1.01·22-s − 0.379·23-s − 0.0111·24-s − 0.922·25-s + 0.909·26-s + 0.0632·27-s + 0.296·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6002\)    =    \(2 \cdot 3001\)
Sign: $-1$
Analytic conductor: \(47.9262\)
Root analytic conductor: \(6.92287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6002,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3001 \( 1+O(T) \)
good3 \( 1 + 0.0548T + 3T^{2} \)
5 \( 1 - 0.621T + 5T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 + 4.75T + 11T^{2} \)
13 \( 1 - 4.63T + 13T^{2} \)
17 \( 1 + 4.97T + 17T^{2} \)
19 \( 1 + 2.09T + 19T^{2} \)
23 \( 1 + 1.82T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 4.13T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 3.20T + 41T^{2} \)
43 \( 1 + 7.28T + 43T^{2} \)
47 \( 1 - 2.07T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 + 4.77T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 7.62T + 67T^{2} \)
71 \( 1 + 5.48T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.916584369469429218171876046315, −6.68721750720519948177626156655, −6.29310998081928857335567808529, −5.45562500917260344902424316621, −4.94863401771645831473999165123, −4.14908220592806725493918020242, −3.15743928339173778941789725434, −2.47093419304656375556455301018, −1.61021325103828935206842305554, 0, 1.61021325103828935206842305554, 2.47093419304656375556455301018, 3.15743928339173778941789725434, 4.14908220592806725493918020242, 4.94863401771645831473999165123, 5.45562500917260344902424316621, 6.29310998081928857335567808529, 6.68721750720519948177626156655, 7.916584369469429218171876046315

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