Properties

Label 2-2475-1.1-c3-0-141
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $146.029$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.66·2-s + 13.7·4-s − 20.4·7-s − 26.6·8-s + 11·11-s + 75.4·13-s + 95.1·14-s + 14.4·16-s + 57.7·17-s − 27.5·19-s − 51.2·22-s + 125.·23-s − 351.·26-s − 279.·28-s − 73.6·29-s − 316.·31-s + 145.·32-s − 268.·34-s − 71.5·37-s + 128.·38-s − 359.·41-s − 380.·43-s + 150.·44-s − 585.·46-s + 576.·47-s + 73.4·49-s + 1.03e3·52-s + ⋯
L(s)  = 1  − 1.64·2-s + 1.71·4-s − 1.10·7-s − 1.17·8-s + 0.301·11-s + 1.61·13-s + 1.81·14-s + 0.225·16-s + 0.823·17-s − 0.332·19-s − 0.496·22-s + 1.13·23-s − 2.65·26-s − 1.88·28-s − 0.471·29-s − 1.83·31-s + 0.805·32-s − 1.35·34-s − 0.318·37-s + 0.547·38-s − 1.36·41-s − 1.34·43-s + 0.517·44-s − 1.87·46-s + 1.78·47-s + 0.214·49-s + 2.76·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(146.029\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2475,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 4.66T + 8T^{2} \)
7 \( 1 + 20.4T + 343T^{2} \)
13 \( 1 - 75.4T + 2.19e3T^{2} \)
17 \( 1 - 57.7T + 4.91e3T^{2} \)
19 \( 1 + 27.5T + 6.85e3T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 + 73.6T + 2.43e4T^{2} \)
31 \( 1 + 316.T + 2.97e4T^{2} \)
37 \( 1 + 71.5T + 5.06e4T^{2} \)
41 \( 1 + 359.T + 6.89e4T^{2} \)
43 \( 1 + 380.T + 7.95e4T^{2} \)
47 \( 1 - 576.T + 1.03e5T^{2} \)
53 \( 1 + 252.T + 1.48e5T^{2} \)
59 \( 1 - 552.T + 2.05e5T^{2} \)
61 \( 1 + 9.30T + 2.26e5T^{2} \)
67 \( 1 + 610.T + 3.00e5T^{2} \)
71 \( 1 + 132.T + 3.57e5T^{2} \)
73 \( 1 - 220.T + 3.89e5T^{2} \)
79 \( 1 - 9.99e2T + 4.93e5T^{2} \)
83 \( 1 - 229.T + 5.71e5T^{2} \)
89 \( 1 - 178.T + 7.04e5T^{2} \)
97 \( 1 + 104.T + 9.12e5T^{2} \)
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

−8.482883363434380004609655528373, −7.53903036442406833773555651562, −6.84089423044659847099321138504, −6.26258945861292929571469287933, −5.33986671588517192722339809630, −3.77908289715107635310028291362, −3.17542190912253620478672965185, −1.84972059580280527118611495164, −1.01162658253182613210369896565, 0, 1.01162658253182613210369896565, 1.84972059580280527118611495164, 3.17542190912253620478672965185, 3.77908289715107635310028291362, 5.33986671588517192722339809630, 6.26258945861292929571469287933, 6.84089423044659847099321138504, 7.53903036442406833773555651562, 8.482883363434380004609655528373

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