Properties

Label 2-6017-1.1-c1-0-17
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.47·2-s − 1.34·3-s + 4.11·4-s + 2.94·5-s + 3.32·6-s − 2.85·7-s − 5.23·8-s − 1.18·9-s − 7.28·10-s − 11-s − 5.54·12-s − 1.72·13-s + 7.05·14-s − 3.96·15-s + 4.71·16-s − 6.02·17-s + 2.93·18-s − 4.89·19-s + 12.1·20-s + 3.83·21-s + 2.47·22-s + 3.06·23-s + 7.04·24-s + 3.68·25-s + 4.27·26-s + 5.63·27-s − 11.7·28-s + ⋯
L(s)  = 1  − 1.74·2-s − 0.777·3-s + 2.05·4-s + 1.31·5-s + 1.35·6-s − 1.07·7-s − 1.85·8-s − 0.396·9-s − 2.30·10-s − 0.301·11-s − 1.59·12-s − 0.479·13-s + 1.88·14-s − 1.02·15-s + 1.17·16-s − 1.46·17-s + 0.692·18-s − 1.12·19-s + 2.71·20-s + 0.837·21-s + 0.527·22-s + 0.639·23-s + 1.43·24-s + 0.736·25-s + 0.838·26-s + 1.08·27-s − 2.21·28-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

\(L(1)\) \(\approx\) \(0.1136552488\)
\(L(\frac12)\) \(\approx\) \(0.1136552488\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.47T + 2T^{2} \)
3 \( 1 + 1.34T + 3T^{2} \)
5 \( 1 - 2.94T + 5T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
13 \( 1 + 1.72T + 13T^{2} \)
17 \( 1 + 6.02T + 17T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 + 9.51T + 29T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + 2.03T + 37T^{2} \)
41 \( 1 + 2.59T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 4.16T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 4.62T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 7.80T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 8.61T + 79T^{2} \)
83 \( 1 + 8.70T + 83T^{2} \)
89 \( 1 - 7.51T + 89T^{2} \)
97 \( 1 + 3.08T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.381732961662071720613174643096, −7.22155543119067777019426625048, −6.74388654479056455663611114703, −6.19123012678162835595943649401, −5.64300628256851572983345429850, −4.69227669182466420332100686756, −3.19024632050134158407706102289, −2.30310092949870673869280631472, −1.73390558817644495411198964980, −0.22600458316666464249232512515, 0.22600458316666464249232512515, 1.73390558817644495411198964980, 2.30310092949870673869280631472, 3.19024632050134158407706102289, 4.69227669182466420332100686756, 5.64300628256851572983345429850, 6.19123012678162835595943649401, 6.74388654479056455663611114703, 7.22155543119067777019426625048, 8.381732961662071720613174643096

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