Properties

Label 2-6002-1.1-c1-0-241
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 + 2.65·3-s + 4-s − 2.01·5-s + 2.65·6-s − 3.04·7-s + 8-s + 4.03·9-s − 2.01·10-s − 0.414·11-s + 2.65·12-s + 0.200·13-s − 3.04·14-s − 5.35·15-s + 16-s − 2.08·17-s + 4.03·18-s − 2.37·19-s − 2.01·20-s − 8.07·21-s − 0.414·22-s + 1.24·23-s + 2.65·24-s − 0.920·25-s + 0.200·26-s + 2.75·27-s − 3.04·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.53·3-s + 0.5·4-s − 0.903·5-s + 1.08·6-s − 1.15·7-s + 0.353·8-s + 1.34·9-s − 0.638·10-s − 0.124·11-s + 0.765·12-s + 0.0556·13-s − 0.813·14-s − 1.38·15-s + 0.250·16-s − 0.505·17-s + 0.951·18-s − 0.544·19-s − 0.451·20-s − 1.76·21-s − 0.0883·22-s + 0.259·23-s + 0.541·24-s − 0.184·25-s + 0.0393·26-s + 0.530·27-s − 0.575·28-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 \( 1 - 2.65T + 3T^{2} \)
5 \( 1 + 2.01T + 5T^{2} \)
7 \( 1 + 3.04T + 7T^{2} \)
11 \( 1 + 0.414T + 11T^{2} \)
13 \( 1 - 0.200T + 13T^{2} \)
17 \( 1 + 2.08T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 - 1.24T + 23T^{2} \)
29 \( 1 + 4.35T + 29T^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 + 1.62T + 37T^{2} \)
41 \( 1 + 6.87T + 41T^{2} \)
43 \( 1 + 9.27T + 43T^{2} \)
47 \( 1 + 7.77T + 47T^{2} \)
53 \( 1 + 0.0989T + 53T^{2} \)
59 \( 1 - 4.70T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 - 9.09T + 67T^{2} \)
71 \( 1 + 6.36T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 9.34T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + 5.45T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−7.79237323719240989993021694609, −6.94734186664294241516793387691, −6.59957086844625303329880116984, −5.49380754629961722578154460731, −4.49864192509634095730937297937, −3.75850601454906675144793307252, −3.37661851910510909543904303404, −2.65837789499398366845978466487, −1.77579129592883001138024986772, 0, 1.77579129592883001138024986772, 2.65837789499398366845978466487, 3.37661851910510909543904303404, 3.75850601454906675144793307252, 4.49864192509634095730937297937, 5.49380754629961722578154460731, 6.59957086844625303329880116984, 6.94734186664294241516793387691, 7.79237323719240989993021694609

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