Properties

Label 2-7800-1.1-c1-0-96
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
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  + 3-s + 9-s − 2.86·11-s + 13-s − 5.52·17-s + 3.52·19-s − 7.52·23-s + 27-s + 6.77·29-s + 5.72·31-s − 2.86·33-s + 3.72·37-s + 39-s − 10.1·41-s + 5.52·43-s − 8.65·47-s − 7·49-s − 5.52·51-s + 6.77·53-s + 3.52·57-s + 0.593·59-s − 5.25·61-s + 10.5·67-s − 7.52·69-s − 2.38·71-s − 5.45·73-s + 2.47·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.333·9-s − 0.863·11-s + 0.277·13-s − 1.33·17-s + 0.808·19-s − 1.56·23-s + 0.192·27-s + 1.25·29-s + 1.02·31-s − 0.498·33-s + 0.613·37-s + 0.160·39-s − 1.58·41-s + 0.842·43-s − 1.26·47-s − 49-s − 0.773·51-s + 0.930·53-s + 0.466·57-s + 0.0773·59-s − 0.672·61-s + 1.28·67-s − 0.905·69-s − 0.283·71-s − 0.638·73-s + 0.278·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 2.86T + 11T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 - 3.52T + 19T^{2} \)
23 \( 1 + 7.52T + 23T^{2} \)
29 \( 1 - 6.77T + 29T^{2} \)
31 \( 1 - 5.72T + 31T^{2} \)
37 \( 1 - 3.72T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 5.52T + 43T^{2} \)
47 \( 1 + 8.65T + 47T^{2} \)
53 \( 1 - 6.77T + 53T^{2} \)
59 \( 1 - 0.593T + 59T^{2} \)
61 \( 1 + 5.25T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 2.38T + 71T^{2} \)
73 \( 1 + 5.45T + 73T^{2} \)
79 \( 1 - 2.47T + 79T^{2} \)
83 \( 1 + 8.11T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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

−7.67215588937917696884675765736, −6.77953939702828842126594776782, −6.27699128499688535294274464680, −5.33471249725517386297250747019, −4.60479022217698224353097100293, −3.94130844640860387594086373671, −2.95804649553138258896601839949, −2.39642490723420324740181388671, −1.38887103708305694508247303011, 0, 1.38887103708305694508247303011, 2.39642490723420324740181388671, 2.95804649553138258896601839949, 3.94130844640860387594086373671, 4.60479022217698224353097100293, 5.33471249725517386297250747019, 6.27699128499688535294274464680, 6.77953939702828842126594776782, 7.67215588937917696884675765736

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