Properties

Label 2-6552-1.1-c1-0-67
Degree $2$
Conductor $6552$
Sign $-1$
Analytic cond. $52.3179$
Root an. cond. $7.23311$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.37·5-s − 7-s − 5.37·11-s − 13-s + 5.37·17-s + 7.37·19-s − 1.37·23-s − 3.11·25-s − 4.62·29-s − 10.7·31-s − 1.37·35-s + 5.37·37-s + 6·41-s + 7.37·43-s − 9.48·47-s + 49-s + 2.74·53-s − 7.37·55-s + 2.74·59-s − 12.1·61-s − 1.37·65-s + 12·67-s − 10·71-s + 9.37·73-s + 5.37·77-s − 14.7·79-s + 1.25·83-s + ⋯
L(s)  = 1  + 0.613·5-s − 0.377·7-s − 1.61·11-s − 0.277·13-s + 1.30·17-s + 1.69·19-s − 0.286·23-s − 0.623·25-s − 0.859·29-s − 1.92·31-s − 0.231·35-s + 0.883·37-s + 0.937·41-s + 1.12·43-s − 1.38·47-s + 0.142·49-s + 0.376·53-s − 0.994·55-s + 0.357·59-s − 1.55·61-s − 0.170·65-s + 1.46·67-s − 1.18·71-s + 1.09·73-s + 0.612·77-s − 1.65·79-s + 0.137·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6552\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(52.3179\)
Root analytic conductor: \(7.23311\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6552,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 - 1.37T + 5T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 - 7.37T + 19T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 + 4.62T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 5.37T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 7.37T + 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 - 1.25T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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

−7.55427909697988221866632301963, −7.24652779700113212882652137575, −5.88625418611655523484049834569, −5.63154240607427381068309079110, −5.06285038021067423666291664724, −3.88080275562419193051011418749, −3.08411698014376635450029956947, −2.40161638388589515391796808353, −1.33994444912205386639549027614, 0, 1.33994444912205386639549027614, 2.40161638388589515391796808353, 3.08411698014376635450029956947, 3.88080275562419193051011418749, 5.06285038021067423666291664724, 5.63154240607427381068309079110, 5.88625418611655523484049834569, 7.24652779700113212882652137575, 7.55427909697988221866632301963

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