Properties

Label 2-9075-1.1-c1-0-125
Degree $2$
Conductor $9075$
Sign $1$
Analytic cond. $72.4642$
Root an. cond. $8.51259$
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  − 2.21·2-s + 3-s + 2.90·4-s − 2.21·6-s + 1.59·7-s − 1.99·8-s + 9-s + 2.90·12-s + 0.891·13-s − 3.52·14-s − 1.38·16-s + 4.43·17-s − 2.21·18-s + 5.84·19-s + 1.59·21-s − 3.27·23-s − 1.99·24-s − 1.97·26-s + 27-s + 4.62·28-s − 5.30·29-s + 8.91·31-s + 7.05·32-s − 9.82·34-s + 2.90·36-s − 5.53·37-s − 12.9·38-s + ⋯
L(s)  = 1  − 1.56·2-s + 0.577·3-s + 1.45·4-s − 0.903·6-s + 0.602·7-s − 0.705·8-s + 0.333·9-s + 0.837·12-s + 0.247·13-s − 0.942·14-s − 0.346·16-s + 1.07·17-s − 0.521·18-s + 1.34·19-s + 0.347·21-s − 0.682·23-s − 0.407·24-s − 0.387·26-s + 0.192·27-s + 0.873·28-s − 0.984·29-s + 1.60·31-s + 1.24·32-s − 1.68·34-s + 0.483·36-s − 0.909·37-s − 2.09·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9075\)    =    \(3 \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(72.4642\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.474011900\)
\(L(\frac12)\) \(\approx\) \(1.474011900\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good2 \( 1 + 2.21T + 2T^{2} \)
7 \( 1 - 1.59T + 7T^{2} \)
13 \( 1 - 0.891T + 13T^{2} \)
17 \( 1 - 4.43T + 17T^{2} \)
19 \( 1 - 5.84T + 19T^{2} \)
23 \( 1 + 3.27T + 23T^{2} \)
29 \( 1 + 5.30T + 29T^{2} \)
31 \( 1 - 8.91T + 31T^{2} \)
37 \( 1 + 5.53T + 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 - 6.39T + 43T^{2} \)
47 \( 1 - 8.47T + 47T^{2} \)
53 \( 1 + 4.38T + 53T^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 + 2.03T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 1.15T + 71T^{2} \)
73 \( 1 - 6.66T + 73T^{2} \)
79 \( 1 - 4.89T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + 8.06T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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

−7.905510039306772414409477308821, −7.48225918661961780129469210555, −6.71453264017602272065993641838, −5.82462376697997752933262584982, −5.03237156023926988003180056483, −4.08095323322695892617085356095, −3.20473772839856446995294204038, −2.33375071125182132503517474628, −1.48143669321272040132998553771, −0.792534861731435298038278658154, 0.792534861731435298038278658154, 1.48143669321272040132998553771, 2.33375071125182132503517474628, 3.20473772839856446995294204038, 4.08095323322695892617085356095, 5.03237156023926988003180056483, 5.82462376697997752933262584982, 6.71453264017602272065993641838, 7.48225918661961780129469210555, 7.905510039306772414409477308821

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