Properties

Label 2-6017-1.1-c1-0-40
Degree $2$
Conductor $6017$
Sign $1$
Analytic cond. $48.0459$
Root an. cond. $6.93152$
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  − 2.25·2-s − 2.77·3-s + 3.08·4-s + 0.288·5-s + 6.26·6-s + 1.78·7-s − 2.45·8-s + 4.70·9-s − 0.649·10-s − 11-s − 8.57·12-s + 5.07·13-s − 4.02·14-s − 0.799·15-s − 0.637·16-s − 2.51·17-s − 10.6·18-s − 2.44·19-s + 0.889·20-s − 4.95·21-s + 2.25·22-s − 7.23·23-s + 6.81·24-s − 4.91·25-s − 11.4·26-s − 4.72·27-s + 5.51·28-s + ⋯
L(s)  = 1  − 1.59·2-s − 1.60·3-s + 1.54·4-s + 0.128·5-s + 2.55·6-s + 0.675·7-s − 0.868·8-s + 1.56·9-s − 0.205·10-s − 0.301·11-s − 2.47·12-s + 1.40·13-s − 1.07·14-s − 0.206·15-s − 0.159·16-s − 0.610·17-s − 2.50·18-s − 0.561·19-s + 0.198·20-s − 1.08·21-s + 0.480·22-s − 1.50·23-s + 1.39·24-s − 0.983·25-s − 2.24·26-s − 0.910·27-s + 1.04·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $1$
Analytic conductor: \(48.0459\)
Root analytic conductor: \(6.93152\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6017,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2606546118\)
\(L(\frac12)\) \(\approx\) \(0.2606546118\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
547 \( 1 - T \)
good2 \( 1 + 2.25T + 2T^{2} \)
3 \( 1 + 2.77T + 3T^{2} \)
5 \( 1 - 0.288T + 5T^{2} \)
7 \( 1 - 1.78T + 7T^{2} \)
13 \( 1 - 5.07T + 13T^{2} \)
17 \( 1 + 2.51T + 17T^{2} \)
19 \( 1 + 2.44T + 19T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 + 7.48T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 - 7.61T + 43T^{2} \)
47 \( 1 + 9.12T + 47T^{2} \)
53 \( 1 + 8.32T + 53T^{2} \)
59 \( 1 - 3.22T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 4.64T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 0.0153T + 79T^{2} \)
83 \( 1 - 1.08T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + 8.81T + 97T^{2} \)
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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

−8.193964828623507730888358004857, −7.43632809048946002427684124281, −6.74766253229966475274855382628, −5.99206797568954720392662386084, −5.61667351589242879251602285157, −4.54775937043622716427900568152, −3.79631006313191177746729378364, −2.03020465977158906547472729774, −1.56082989382281963851004547209, −0.38343966772273819351003821141, 0.38343966772273819351003821141, 1.56082989382281963851004547209, 2.03020465977158906547472729774, 3.79631006313191177746729378364, 4.54775937043622716427900568152, 5.61667351589242879251602285157, 5.99206797568954720392662386084, 6.74766253229966475274855382628, 7.43632809048946002427684124281, 8.193964828623507730888358004857

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