Properties

Label 2-1575-1.1-c3-0-71
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  − 1.70·2-s − 5.10·4-s − 7·7-s + 22.2·8-s − 37.4·11-s − 29.0·13-s + 11.9·14-s + 2.89·16-s + 58.4·17-s − 54.5·19-s + 63.6·22-s + 161.·23-s + 49.3·26-s + 35.7·28-s − 137.·29-s + 154.·31-s − 183.·32-s − 99.4·34-s + 350.·37-s + 92.8·38-s − 353.·41-s + 518.·43-s + 190.·44-s − 275.·46-s − 542.·47-s + 49·49-s + 148.·52-s + ⋯
L(s)  = 1  − 0.601·2-s − 0.638·4-s − 0.377·7-s + 0.985·8-s − 1.02·11-s − 0.619·13-s + 0.227·14-s + 0.0452·16-s + 0.833·17-s − 0.659·19-s + 0.616·22-s + 1.46·23-s + 0.372·26-s + 0.241·28-s − 0.880·29-s + 0.896·31-s − 1.01·32-s − 0.501·34-s + 1.55·37-s + 0.396·38-s − 1.34·41-s + 1.83·43-s + 0.654·44-s − 0.881·46-s − 1.68·47-s + 0.142·49-s + 0.394·52-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 \( 1 + 1.70T + 8T^{2} \)
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

−8.761543260320092012523303247857, −7.84234479147763474799855470839, −7.41779908318893452241332078150, −6.25544222955419009800857274325, −5.21806612470637396149750558220, −4.62089319860350269156947609330, −3.44510499800316644381209374912, −2.42003280597919928770959819897, −1.00965334270187999610832995518, 0, 1.00965334270187999610832995518, 2.42003280597919928770959819897, 3.44510499800316644381209374912, 4.62089319860350269156947609330, 5.21806612470637396149750558220, 6.25544222955419009800857274325, 7.41779908318893452241332078150, 7.84234479147763474799855470839, 8.761543260320092012523303247857

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