Properties

Label 2-6002-1.1-c1-0-170
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.82·3-s + 4-s − 1.67·5-s − 2.82·6-s + 4.44·7-s + 8-s + 4.97·9-s − 1.67·10-s − 0.291·11-s − 2.82·12-s − 1.11·13-s + 4.44·14-s + 4.74·15-s + 16-s + 6.41·17-s + 4.97·18-s − 6.21·19-s − 1.67·20-s − 12.5·21-s − 0.291·22-s − 0.0504·23-s − 2.82·24-s − 2.17·25-s − 1.11·26-s − 5.58·27-s + 4.44·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.63·3-s + 0.5·4-s − 0.751·5-s − 1.15·6-s + 1.67·7-s + 0.353·8-s + 1.65·9-s − 0.531·10-s − 0.0879·11-s − 0.815·12-s − 0.309·13-s + 1.18·14-s + 1.22·15-s + 0.250·16-s + 1.55·17-s + 1.17·18-s − 1.42·19-s − 0.375·20-s − 2.73·21-s − 0.0621·22-s − 0.0105·23-s − 0.576·24-s − 0.435·25-s − 0.218·26-s − 1.07·27-s + 0.839·28-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 \( 1 + 2.82T + 3T^{2} \)
5 \( 1 + 1.67T + 5T^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 + 0.291T + 11T^{2} \)
13 \( 1 + 1.11T + 13T^{2} \)
17 \( 1 - 6.41T + 17T^{2} \)
19 \( 1 + 6.21T + 19T^{2} \)
23 \( 1 + 0.0504T + 23T^{2} \)
29 \( 1 + 8.58T + 29T^{2} \)
31 \( 1 + 4.85T + 31T^{2} \)
37 \( 1 + 8.46T + 37T^{2} \)
41 \( 1 + 5.07T + 41T^{2} \)
43 \( 1 - 9.67T + 43T^{2} \)
47 \( 1 - 3.87T + 47T^{2} \)
53 \( 1 - 1.70T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 7.46T + 67T^{2} \)
71 \( 1 + 4.96T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−7.59799257628500848190299906931, −6.99686530209414998915470160665, −6.00621322369330669032335219575, −5.40328082468153544673392840112, −5.04351216647120204008086997647, −4.20473362846778686844756569848, −3.70099487525356697028701592511, −2.11800213366736148669890975529, −1.30518553417661462992189214866, 0, 1.30518553417661462992189214866, 2.11800213366736148669890975529, 3.70099487525356697028701592511, 4.20473362846778686844756569848, 5.04351216647120204008086997647, 5.40328082468153544673392840112, 6.00621322369330669032335219575, 6.99686530209414998915470160665, 7.59799257628500848190299906931

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