Properties

Label 2-6026-1.1-c1-0-11
Degree $2$
Conductor $6026$
Sign $1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
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-s − 2.51·3-s + 4-s − 0.511·5-s + 2.51·6-s + 2.88·7-s − 8-s + 3.30·9-s + 0.511·10-s − 5.38·11-s − 2.51·12-s + 3.75·13-s − 2.88·14-s + 1.28·15-s + 16-s − 6.96·17-s − 3.30·18-s − 8.21·19-s − 0.511·20-s − 7.24·21-s + 5.38·22-s − 23-s + 2.51·24-s − 4.73·25-s − 3.75·26-s − 0.760·27-s + 2.88·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.44·3-s + 0.5·4-s − 0.228·5-s + 1.02·6-s + 1.09·7-s − 0.353·8-s + 1.10·9-s + 0.161·10-s − 1.62·11-s − 0.724·12-s + 1.04·13-s − 0.771·14-s + 0.331·15-s + 0.250·16-s − 1.68·17-s − 0.778·18-s − 1.88·19-s − 0.114·20-s − 1.58·21-s + 1.14·22-s − 0.208·23-s + 0.512·24-s − 0.947·25-s − 0.736·26-s − 0.146·27-s + 0.545·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6026\)    =    \(2 \cdot 23 \cdot 131\)
Sign: $1$
Analytic conductor: \(48.1178\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2004542437\)
\(L(\frac12)\) \(\approx\) \(0.2004542437\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 + T \)
131 \( 1 - T \)
good3 \( 1 + 2.51T + 3T^{2} \)
5 \( 1 + 0.511T + 5T^{2} \)
7 \( 1 - 2.88T + 7T^{2} \)
11 \( 1 + 5.38T + 11T^{2} \)
13 \( 1 - 3.75T + 13T^{2} \)
17 \( 1 + 6.96T + 17T^{2} \)
19 \( 1 + 8.21T + 19T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 + 5.58T + 31T^{2} \)
37 \( 1 + 3.38T + 37T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + 7.08T + 47T^{2} \)
53 \( 1 - 7.83T + 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + 2.01T + 71T^{2} \)
73 \( 1 - 2.53T + 73T^{2} \)
79 \( 1 - 1.71T + 79T^{2} \)
83 \( 1 + 2.22T + 83T^{2} \)
89 \( 1 - 8.71T + 89T^{2} \)
97 \( 1 + 9.11T + 97T^{2} \)
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

−8.198822504336426156975954777309, −7.37666846386337787839409896648, −6.64174145034245096563746543975, −5.93353737309120518034154907583, −5.37809824686459120223100401886, −4.59142641796243070760917632632, −3.89748081529409276916525283522, −2.32301830036827788977391414071, −1.72864211838059409579463171109, −0.27272816084944141917221126478, 0.27272816084944141917221126478, 1.72864211838059409579463171109, 2.32301830036827788977391414071, 3.89748081529409276916525283522, 4.59142641796243070760917632632, 5.37809824686459120223100401886, 5.93353737309120518034154907583, 6.64174145034245096563746543975, 7.37666846386337787839409896648, 8.198822504336426156975954777309

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