Properties

Label 2-6010-1.1-c1-0-70
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 + 3.24·3-s + 4-s + 5-s − 3.24·6-s − 1.95·7-s − 8-s + 7.49·9-s − 10-s − 3.33·11-s + 3.24·12-s − 4.86·13-s + 1.95·14-s + 3.24·15-s + 16-s + 6.16·17-s − 7.49·18-s + 2.86·19-s + 20-s − 6.32·21-s + 3.33·22-s − 0.559·23-s − 3.24·24-s + 25-s + 4.86·26-s + 14.5·27-s − 1.95·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.87·3-s + 0.5·4-s + 0.447·5-s − 1.32·6-s − 0.737·7-s − 0.353·8-s + 2.49·9-s − 0.316·10-s − 1.00·11-s + 0.935·12-s − 1.34·13-s + 0.521·14-s + 0.836·15-s + 0.250·16-s + 1.49·17-s − 1.76·18-s + 0.656·19-s + 0.223·20-s − 1.38·21-s + 0.711·22-s − 0.116·23-s − 0.661·24-s + 0.200·25-s + 0.954·26-s + 2.80·27-s − 0.368·28-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

\(L(1)\) \(\approx\) \(2.933200945\)
\(L(\frac12)\) \(\approx\) \(2.933200945\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.24T + 3T^{2} \)
7 \( 1 + 1.95T + 7T^{2} \)
11 \( 1 + 3.33T + 11T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 - 6.16T + 17T^{2} \)
19 \( 1 - 2.86T + 19T^{2} \)
23 \( 1 + 0.559T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + 7.59T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 - 4.97T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 4.98T + 59T^{2} \)
61 \( 1 - 7.51T + 61T^{2} \)
67 \( 1 - 9.04T + 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 + 5.38T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 5.80T + 83T^{2} \)
89 \( 1 + 1.03T + 89T^{2} \)
97 \( 1 + 7.21T + 97T^{2} \)
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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

−7.84478684965780563613554974341, −7.67691006924337074102402618234, −7.20076148371019461374869886338, −6.07032636053383302981757480871, −5.24860410617494531934596396520, −4.17321281333475152336141690565, −3.20168485573193164840672810060, −2.70166442644008976096283382253, −2.14287087322373625475912635365, −0.913307570908189847858473669472, 0.913307570908189847858473669472, 2.14287087322373625475912635365, 2.70166442644008976096283382253, 3.20168485573193164840672810060, 4.17321281333475152336141690565, 5.24860410617494531934596396520, 6.07032636053383302981757480871, 7.20076148371019461374869886338, 7.67691006924337074102402618234, 7.84478684965780563613554974341

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