Properties

Label 2-9555-1.1-c1-0-37
Degree $2$
Conductor $9555$
Sign $1$
Analytic cond. $76.2970$
Root an. cond. $8.73481$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 9-s − 2·10-s − 5·11-s − 2·12-s − 13-s + 15-s − 4·16-s − 5·17-s + 2·18-s − 2·19-s − 2·20-s − 10·22-s − 23-s + 25-s − 2·26-s − 27-s + 10·29-s + 2·30-s + 2·31-s − 8·32-s + 5·33-s − 10·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 1.50·11-s − 0.577·12-s − 0.277·13-s + 0.258·15-s − 16-s − 1.21·17-s + 0.471·18-s − 0.458·19-s − 0.447·20-s − 2.13·22-s − 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.85·29-s + 0.365·30-s + 0.359·31-s − 1.41·32-s + 0.870·33-s − 1.71·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9555\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(76.2970\)
Root analytic conductor: \(8.73481\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9555,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774223212\)
\(L(\frac12)\) \(\approx\) \(1.774223212\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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

−7.40964757158743755512563714704, −6.75505898429196133498146650029, −6.19716944730822085541922010748, −5.47260415640564369856793290111, −4.71337262998831410437380055667, −4.56940847343860495087618428417, −3.60242855766606390035844168544, −2.73730155068337513795522381030, −2.17440183294258690818518737075, −0.49221325598293311557199906729, 0.49221325598293311557199906729, 2.17440183294258690818518737075, 2.73730155068337513795522381030, 3.60242855766606390035844168544, 4.56940847343860495087618428417, 4.71337262998831410437380055667, 5.47260415640564369856793290111, 6.19716944730822085541922010748, 6.75505898429196133498146650029, 7.40964757158743755512563714704

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