Properties

Label 2-6010-1.1-c1-0-172
Degree $2$
Conductor $6010$
Sign $-1$
Analytic cond. $47.9900$
Root an. cond. $6.92748$
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 − 1.30·3-s + 4-s − 5-s − 1.30·6-s + 4.54·7-s + 8-s − 1.28·9-s − 10-s + 1.13·11-s − 1.30·12-s + 1.30·13-s + 4.54·14-s + 1.30·15-s + 16-s − 5.70·17-s − 1.28·18-s − 0.0584·19-s − 20-s − 5.94·21-s + 1.13·22-s − 3.59·23-s − 1.30·24-s + 25-s + 1.30·26-s + 5.60·27-s + 4.54·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.755·3-s + 0.5·4-s − 0.447·5-s − 0.533·6-s + 1.71·7-s + 0.353·8-s − 0.429·9-s − 0.316·10-s + 0.342·11-s − 0.377·12-s + 0.361·13-s + 1.21·14-s + 0.337·15-s + 0.250·16-s − 1.38·17-s − 0.304·18-s − 0.0134·19-s − 0.223·20-s − 1.29·21-s + 0.241·22-s − 0.749·23-s − 0.266·24-s + 0.200·25-s + 0.255·26-s + 1.07·27-s + 0.859·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6010\)    =    \(2 \cdot 5 \cdot 601\)
Sign: $-1$
Analytic conductor: \(47.9900\)
Root analytic conductor: \(6.92748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6010,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
601 \( 1 - T \)
good3 \( 1 + 1.30T + 3T^{2} \)
7 \( 1 - 4.54T + 7T^{2} \)
11 \( 1 - 1.13T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 + 0.0584T + 19T^{2} \)
23 \( 1 + 3.59T + 23T^{2} \)
29 \( 1 + 8.50T + 29T^{2} \)
31 \( 1 + 0.0788T + 31T^{2} \)
37 \( 1 + 6.71T + 37T^{2} \)
41 \( 1 + 6.64T + 41T^{2} \)
43 \( 1 + 9.54T + 43T^{2} \)
47 \( 1 + 9.37T + 47T^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 - 6.23T + 67T^{2} \)
71 \( 1 + 2.26T + 71T^{2} \)
73 \( 1 - 7.12T + 73T^{2} \)
79 \( 1 + 6.46T + 79T^{2} \)
83 \( 1 + 5.70T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 6.04T + 97T^{2} \)
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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

−7.65646517411972231042683938147, −6.83776526975295743174211281081, −6.25607357043010245300481223715, −5.26270132564272615620638684119, −5.03764791416549581154197703915, −4.18208204092625794433766723619, −3.50790345855463315533246248046, −2.19020223206050195227249382421, −1.51592712745743956736224636748, 0, 1.51592712745743956736224636748, 2.19020223206050195227249382421, 3.50790345855463315533246248046, 4.18208204092625794433766723619, 5.03764791416549581154197703915, 5.26270132564272615620638684119, 6.25607357043010245300481223715, 6.83776526975295743174211281081, 7.65646517411972231042683938147

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