Properties

Label 2-525-1.1-c3-0-39
Degree $2$
Conductor $525$
Sign $-1$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
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  − 3.56·2-s + 3·3-s + 4.68·4-s − 10.6·6-s + 7·7-s + 11.8·8-s + 9·9-s − 5.19·11-s + 14.0·12-s − 54.5·13-s − 24.9·14-s − 79.5·16-s − 16.1·17-s − 32.0·18-s + 87.4·19-s + 21·21-s + 18.4·22-s − 176.·23-s + 35.4·24-s + 194.·26-s + 27·27-s + 32.7·28-s + 142.·29-s − 94.3·31-s + 188.·32-s − 15.5·33-s + 57.5·34-s + ⋯
L(s)  = 1  − 1.25·2-s + 0.577·3-s + 0.585·4-s − 0.726·6-s + 0.377·7-s + 0.521·8-s + 0.333·9-s − 0.142·11-s + 0.338·12-s − 1.16·13-s − 0.475·14-s − 1.24·16-s − 0.230·17-s − 0.419·18-s + 1.05·19-s + 0.218·21-s + 0.179·22-s − 1.59·23-s + 0.301·24-s + 1.46·26-s + 0.192·27-s + 0.221·28-s + 0.910·29-s − 0.546·31-s + 1.04·32-s − 0.0821·33-s + 0.290·34-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: $-1$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -1)\)

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 \( 1 - 3T \)
5 \( 1 \)
7 \( 1 - 7T \)
good2 \( 1 + 3.56T + 8T^{2} \)
11 \( 1 + 5.19T + 1.33e3T^{2} \)
13 \( 1 + 54.5T + 2.19e3T^{2} \)
17 \( 1 + 16.1T + 4.91e3T^{2} \)
19 \( 1 - 87.4T + 6.85e3T^{2} \)
23 \( 1 + 176.T + 1.21e4T^{2} \)
29 \( 1 - 142.T + 2.43e4T^{2} \)
31 \( 1 + 94.3T + 2.97e4T^{2} \)
37 \( 1 + 17.3T + 5.06e4T^{2} \)
41 \( 1 - 210.T + 6.89e4T^{2} \)
43 \( 1 + 521.T + 7.95e4T^{2} \)
47 \( 1 - 105.T + 1.03e5T^{2} \)
53 \( 1 - 108.T + 1.48e5T^{2} \)
59 \( 1 - 210.T + 2.05e5T^{2} \)
61 \( 1 + 674.T + 2.26e5T^{2} \)
67 \( 1 + 324.T + 3.00e5T^{2} \)
71 \( 1 - 793.T + 3.57e5T^{2} \)
73 \( 1 + 315.T + 3.89e5T^{2} \)
79 \( 1 + 425.T + 4.93e5T^{2} \)
83 \( 1 - 283.T + 5.71e5T^{2} \)
89 \( 1 + 843.T + 7.04e5T^{2} \)
97 \( 1 + 1.53e3T + 9.12e5T^{2} \)
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

−9.849708628900840625392888038927, −9.182752734382268522602132133826, −8.189074437725777677509591503270, −7.69531193756662675881230856221, −6.80038337320182259248395398509, −5.24431720393011742867523080898, −4.17748629582954838913062794466, −2.61924209101489875657644639319, −1.50634273801803250279664283532, 0, 1.50634273801803250279664283532, 2.61924209101489875657644639319, 4.17748629582954838913062794466, 5.24431720393011742867523080898, 6.80038337320182259248395398509, 7.69531193756662675881230856221, 8.189074437725777677509591503270, 9.182752734382268522602132133826, 9.849708628900840625392888038927

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