Properties

Label 2-7800-1.1-c1-0-15
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.36·7-s + 9-s − 5.64·11-s − 13-s + 2.14·17-s + 2.41·19-s − 1.36·21-s − 2.72·23-s + 27-s − 5.28·29-s − 1.64·31-s − 5.64·33-s + 3.86·37-s − 39-s + 5.86·41-s − 9.00·43-s + 3.77·47-s − 5.14·49-s + 2.14·51-s + 12.0·53-s + 2.41·57-s + 6.19·59-s + 11.5·61-s − 1.36·63-s − 0.324·67-s − 2.72·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.515·7-s + 0.333·9-s − 1.70·11-s − 0.277·13-s + 0.519·17-s + 0.553·19-s − 0.297·21-s − 0.568·23-s + 0.192·27-s − 0.980·29-s − 0.295·31-s − 0.982·33-s + 0.635·37-s − 0.160·39-s + 0.916·41-s − 1.37·43-s + 0.551·47-s − 0.734·49-s + 0.299·51-s + 1.64·53-s + 0.319·57-s + 0.806·59-s + 1.48·61-s − 0.171·63-s − 0.0396·67-s − 0.328·69-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: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.769474786\)
\(L(\frac12)\) \(\approx\) \(1.769474786\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.36T + 7T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
17 \( 1 - 2.14T + 17T^{2} \)
19 \( 1 - 2.41T + 19T^{2} \)
23 \( 1 + 2.72T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 - 3.86T + 37T^{2} \)
41 \( 1 - 5.86T + 41T^{2} \)
43 \( 1 + 9.00T + 43T^{2} \)
47 \( 1 - 3.77T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 - 6.19T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 0.324T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 8.28T + 73T^{2} \)
79 \( 1 - 2.31T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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

−7.961747712010711864184296710845, −7.28732819317045322710340979068, −6.61487731310417875338173927364, −5.47864846213836003309431401787, −5.32203197887047253534596418842, −4.15295630319403242015120352751, −3.43568263029269544148650132176, −2.67104844372336206001247091008, −2.03533605584921324542492607317, −0.61467378067749353879409291721, 0.61467378067749353879409291721, 2.03533605584921324542492607317, 2.67104844372336206001247091008, 3.43568263029269544148650132176, 4.15295630319403242015120352751, 5.32203197887047253534596418842, 5.47864846213836003309431401787, 6.61487731310417875338173927364, 7.28732819317045322710340979068, 7.961747712010711864184296710845

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