Properties

Label 2-1575-1.1-c3-0-86
Degree $2$
Conductor $1575$
Sign $-1$
Analytic cond. $92.9280$
Root an. cond. $9.63991$
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  − 1.19·2-s − 6.56·4-s − 7·7-s + 17.4·8-s + 4.52·11-s + 14.5·13-s + 8.37·14-s + 31.7·16-s + 15.5·17-s − 14.8·19-s − 5.41·22-s − 166.·23-s − 17.4·26-s + 45.9·28-s − 194.·29-s + 104.·31-s − 177.·32-s − 18.6·34-s − 47.5·37-s + 17.7·38-s + 378.·41-s + 194.·43-s − 29.7·44-s + 199.·46-s + 488.·47-s + 49·49-s − 95.7·52-s + ⋯
L(s)  = 1  − 0.422·2-s − 0.821·4-s − 0.377·7-s + 0.770·8-s + 0.124·11-s + 0.310·13-s + 0.159·14-s + 0.495·16-s + 0.221·17-s − 0.179·19-s − 0.0524·22-s − 1.51·23-s − 0.131·26-s + 0.310·28-s − 1.24·29-s + 0.605·31-s − 0.979·32-s − 0.0938·34-s − 0.211·37-s + 0.0758·38-s + 1.44·41-s + 0.691·43-s − 0.101·44-s + 0.639·46-s + 1.51·47-s + 0.142·49-s − 0.255·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(92.9280\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1575,\ (\ :3/2),\ -1)\)

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 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 + 1.19T + 8T^{2} \)
11 \( 1 - 4.52T + 1.33e3T^{2} \)
13 \( 1 - 14.5T + 2.19e3T^{2} \)
17 \( 1 - 15.5T + 4.91e3T^{2} \)
19 \( 1 + 14.8T + 6.85e3T^{2} \)
23 \( 1 + 166.T + 1.21e4T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 - 104.T + 2.97e4T^{2} \)
37 \( 1 + 47.5T + 5.06e4T^{2} \)
41 \( 1 - 378.T + 6.89e4T^{2} \)
43 \( 1 - 194.T + 7.95e4T^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 - 316.T + 1.48e5T^{2} \)
59 \( 1 - 350.T + 2.05e5T^{2} \)
61 \( 1 - 115.T + 2.26e5T^{2} \)
67 \( 1 + 434.T + 3.00e5T^{2} \)
71 \( 1 + 410.T + 3.57e5T^{2} \)
73 \( 1 - 464.T + 3.89e5T^{2} \)
79 \( 1 + 290.T + 4.93e5T^{2} \)
83 \( 1 - 578.T + 5.71e5T^{2} \)
89 \( 1 + 937.T + 7.04e5T^{2} \)
97 \( 1 + 839.T + 9.12e5T^{2} \)
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

−8.791218242302896855792676075115, −7.939613357284127272375508495301, −7.29805054337327390389566662186, −6.10702880066989189109128303553, −5.47363517113385842461577051025, −4.25495826248723634826451929876, −3.75373105645356305031978132027, −2.35800383015989326197550826542, −1.07983027148695969248977662005, 0, 1.07983027148695969248977662005, 2.35800383015989326197550826542, 3.75373105645356305031978132027, 4.25495826248723634826451929876, 5.47363517113385842461577051025, 6.10702880066989189109128303553, 7.29805054337327390389566662186, 7.939613357284127272375508495301, 8.791218242302896855792676075115

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