Properties

Label 2-6010-1.1-c1-0-76
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.61·3-s + 4-s + 5-s + 1.61·6-s − 4.97·7-s + 8-s − 0.390·9-s + 10-s + 2.05·11-s + 1.61·12-s + 6.01·13-s − 4.97·14-s + 1.61·15-s + 16-s − 6.28·17-s − 0.390·18-s + 6.01·19-s + 20-s − 8.02·21-s + 2.05·22-s + 2.50·23-s + 1.61·24-s + 25-s + 6.01·26-s − 5.47·27-s − 4.97·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.932·3-s + 0.5·4-s + 0.447·5-s + 0.659·6-s − 1.87·7-s + 0.353·8-s − 0.130·9-s + 0.316·10-s + 0.620·11-s + 0.466·12-s + 1.66·13-s − 1.32·14-s + 0.417·15-s + 0.250·16-s − 1.52·17-s − 0.0921·18-s + 1.37·19-s + 0.223·20-s − 1.75·21-s + 0.438·22-s + 0.521·23-s + 0.329·24-s + 0.200·25-s + 1.18·26-s − 1.05·27-s − 0.939·28-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: \(0\)
Selberg data: \((2,\ 6010,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.275400901\)
\(L(\frac12)\) \(\approx\) \(4.275400901\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 3T^{2} \)
7 \( 1 + 4.97T + 7T^{2} \)
11 \( 1 - 2.05T + 11T^{2} \)
13 \( 1 - 6.01T + 13T^{2} \)
17 \( 1 + 6.28T + 17T^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 - 2.50T + 23T^{2} \)
29 \( 1 - 2.03T + 29T^{2} \)
31 \( 1 - 0.373T + 31T^{2} \)
37 \( 1 - 3.56T + 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 - 0.00845T + 43T^{2} \)
47 \( 1 + 0.449T + 47T^{2} \)
53 \( 1 - 1.61T + 53T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 - 2.18T + 61T^{2} \)
67 \( 1 + 3.60T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 16.9T + 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 - 1.17T + 83T^{2} \)
89 \( 1 - 8.17T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−8.148584333701613424031459433639, −7.14214871843457477575116522439, −6.44053295714990898756830492039, −6.17722075848719920964957871999, −5.31076258129172258528887781867, −4.07513711228052572761186622164, −3.55571536007184659741970029369, −2.97106466509277232248158780866, −2.25678391192397271625496860139, −0.941667506874337222414320434297, 0.941667506874337222414320434297, 2.25678391192397271625496860139, 2.97106466509277232248158780866, 3.55571536007184659741970029369, 4.07513711228052572761186622164, 5.31076258129172258528887781867, 6.17722075848719920964957871999, 6.44053295714990898756830492039, 7.14214871843457477575116522439, 8.148584333701613424031459433639

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