Properties

Label 2-6026-1.1-c1-0-151
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.936·3-s + 4-s + 1.38·5-s + 0.936·6-s + 2.49·7-s − 8-s − 2.12·9-s − 1.38·10-s − 1.29·11-s − 0.936·12-s − 0.532·13-s − 2.49·14-s − 1.29·15-s + 16-s − 4.49·17-s + 2.12·18-s − 6.56·19-s + 1.38·20-s − 2.33·21-s + 1.29·22-s + 23-s + 0.936·24-s − 3.09·25-s + 0.532·26-s + 4.79·27-s + 2.49·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.540·3-s + 0.5·4-s + 0.617·5-s + 0.382·6-s + 0.942·7-s − 0.353·8-s − 0.707·9-s − 0.436·10-s − 0.390·11-s − 0.270·12-s − 0.147·13-s − 0.666·14-s − 0.334·15-s + 0.250·16-s − 1.08·17-s + 0.500·18-s − 1.50·19-s + 0.308·20-s − 0.509·21-s + 0.276·22-s + 0.208·23-s + 0.191·24-s − 0.618·25-s + 0.104·26-s + 0.923·27-s + 0.471·28-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: \(1\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.936T + 3T^{2} \)
5 \( 1 - 1.38T + 5T^{2} \)
7 \( 1 - 2.49T + 7T^{2} \)
11 \( 1 + 1.29T + 11T^{2} \)
13 \( 1 + 0.532T + 13T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + 6.56T + 19T^{2} \)
29 \( 1 - 9.38T + 29T^{2} \)
31 \( 1 - 8.74T + 31T^{2} \)
37 \( 1 - 5.61T + 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 - 7.17T + 43T^{2} \)
47 \( 1 - 6.97T + 47T^{2} \)
53 \( 1 - 0.607T + 53T^{2} \)
59 \( 1 + 0.436T + 59T^{2} \)
61 \( 1 + 2.24T + 61T^{2} \)
67 \( 1 + 8.74T + 67T^{2} \)
71 \( 1 - 3.67T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + 9.05T + 79T^{2} \)
83 \( 1 + 7.36T + 83T^{2} \)
89 \( 1 + 7.67T + 89T^{2} \)
97 \( 1 + 5.31T + 97T^{2} \)
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

−7.923857932836746890349501553683, −6.96496992978136593881048500021, −6.22726704494184958717026206847, −5.85431334459629639685042662028, −4.78850379481669244458073408103, −4.37602129273523110006669330162, −2.74632354877596414511034921804, −2.30437524516728822387517817983, −1.20674295107536777458686810796, 0, 1.20674295107536777458686810796, 2.30437524516728822387517817983, 2.74632354877596414511034921804, 4.37602129273523110006669330162, 4.78850379481669244458073408103, 5.85431334459629639685042662028, 6.22726704494184958717026206847, 6.96496992978136593881048500021, 7.923857932836746890349501553683

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