Properties

Degree $2$
Conductor $525$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.70·2-s − 3·3-s − 5.10·4-s − 5.10·6-s − 7·7-s − 22.2·8-s + 9·9-s + 37.4·11-s + 15.3·12-s − 29.0·13-s − 11.9·14-s + 2.89·16-s − 58.4·17-s + 15.3·18-s − 54.5·19-s + 21·21-s + 63.6·22-s − 161.·23-s + 66.8·24-s − 49.3·26-s − 27·27-s + 35.7·28-s + 137.·29-s + 154.·31-s + 183.·32-s − 112.·33-s − 99.4·34-s + ⋯
L(s)  = 1  + 0.601·2-s − 0.577·3-s − 0.638·4-s − 0.347·6-s − 0.377·7-s − 0.985·8-s + 0.333·9-s + 1.02·11-s + 0.368·12-s − 0.619·13-s − 0.227·14-s + 0.0452·16-s − 0.833·17-s + 0.200·18-s − 0.659·19-s + 0.218·21-s + 0.616·22-s − 1.46·23-s + 0.568·24-s − 0.372·26-s − 0.192·27-s + 0.241·28-s + 0.880·29-s + 0.896·31-s + 1.01·32-s − 0.591·33-s − 0.501·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.342163959\)
\(L(\frac12)\) \(\approx\) \(1.342163959\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 - 1.70T + 8T^{2} \)
11 \( 1 - 37.4T + 1.33e3T^{2} \)
13 \( 1 + 29.0T + 2.19e3T^{2} \)
17 \( 1 + 58.4T + 4.91e3T^{2} \)
19 \( 1 + 54.5T + 6.85e3T^{2} \)
23 \( 1 + 161.T + 1.21e4T^{2} \)
29 \( 1 - 137.T + 2.43e4T^{2} \)
31 \( 1 - 154.T + 2.97e4T^{2} \)
37 \( 1 - 350.T + 5.06e4T^{2} \)
41 \( 1 - 353.T + 6.89e4T^{2} \)
43 \( 1 - 518.T + 7.95e4T^{2} \)
47 \( 1 - 542.T + 1.03e5T^{2} \)
53 \( 1 + 305.T + 1.48e5T^{2} \)
59 \( 1 - 14.6T + 2.05e5T^{2} \)
61 \( 1 + 171.T + 2.26e5T^{2} \)
67 \( 1 + 551.T + 3.00e5T^{2} \)
71 \( 1 + 120.T + 3.57e5T^{2} \)
73 \( 1 + 284.T + 3.89e5T^{2} \)
79 \( 1 - 941.T + 4.93e5T^{2} \)
83 \( 1 + 377.T + 5.71e5T^{2} \)
89 \( 1 + 677.T + 7.04e5T^{2} \)
97 \( 1 - 1.22e3T + 9.12e5T^{2} \)
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

−10.44952857988875015793601439060, −9.540365068206106608860730546166, −8.873951856178545941469969180437, −7.66450475351796549017430363913, −6.34581754091404579682857301941, −5.94609151761960009021876136681, −4.47848582870770620047149031389, −4.13752717693359713465153361429, −2.55337736460355621813102975690, −0.66253792770971114502580459844, 0.66253792770971114502580459844, 2.55337736460355621813102975690, 4.13752717693359713465153361429, 4.47848582870770620047149031389, 5.94609151761960009021876136681, 6.34581754091404579682857301941, 7.66450475351796549017430363913, 8.873951856178545941469969180437, 9.540365068206106608860730546166, 10.44952857988875015793601439060

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