Properties

Label 2-7800-1.1-c1-0-100
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 − 1.87·7-s + 9-s + 4.15·11-s − 13-s − 17-s − 3.47·19-s − 1.87·21-s + 1.47·23-s + 27-s − 1.80·29-s − 8.96·31-s + 4.15·33-s − 1.72·37-s − 39-s − 8.83·41-s + 4.27·43-s + 1.07·47-s − 3.47·49-s − 51-s + 11.5·53-s − 3.47·57-s − 10.4·59-s − 0.195·61-s − 1.87·63-s − 6.15·67-s + 1.47·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.709·7-s + 0.333·9-s + 1.25·11-s − 0.277·13-s − 0.242·17-s − 0.797·19-s − 0.409·21-s + 0.307·23-s + 0.192·27-s − 0.335·29-s − 1.60·31-s + 0.723·33-s − 0.282·37-s − 0.160·39-s − 1.38·41-s + 0.652·43-s + 0.156·47-s − 0.496·49-s − 0.140·51-s + 1.59·53-s − 0.460·57-s − 1.35·59-s − 0.0250·61-s − 0.236·63-s − 0.752·67-s + 0.177·69-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: \(1\)
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 \( 1 + 1.87T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 3.47T + 19T^{2} \)
23 \( 1 - 1.47T + 23T^{2} \)
29 \( 1 + 1.80T + 29T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + 1.72T + 37T^{2} \)
41 \( 1 + 8.83T + 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 - 1.07T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 0.195T + 61T^{2} \)
67 \( 1 + 6.15T + 67T^{2} \)
71 \( 1 - 4.03T + 71T^{2} \)
73 \( 1 + 1.08T + 73T^{2} \)
79 \( 1 + 8.27T + 79T^{2} \)
83 \( 1 - 4.43T + 83T^{2} \)
89 \( 1 + 1.19T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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

−7.36645298664567748155529582293, −6.89454872223382602174585629331, −6.26822328567150859116702055305, −5.47172987119443798113192483332, −4.48460188821038210185869096401, −3.81650951315804025123480549992, −3.22527335795665915036373892332, −2.23759444606215977270569269316, −1.41888954097044174978271753810, 0, 1.41888954097044174978271753810, 2.23759444606215977270569269316, 3.22527335795665915036373892332, 3.81650951315804025123480549992, 4.48460188821038210185869096401, 5.47172987119443798113192483332, 6.26822328567150859116702055305, 6.89454872223382602174585629331, 7.36645298664567748155529582293

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