Properties

Label 2-1575-1.1-c3-0-113
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  − 5.18·2-s + 18.9·4-s + 7·7-s − 56.5·8-s + 35.9·11-s + 45.2·13-s − 36.3·14-s + 142.·16-s − 113.·17-s + 61.5·19-s − 186.·22-s + 30.6·23-s − 234.·26-s + 132.·28-s − 214.·29-s + 164.·31-s − 284.·32-s + 586.·34-s − 410.·37-s − 319.·38-s + 309.·41-s − 29.9·43-s + 679.·44-s − 158.·46-s − 483.·47-s + 49·49-s + 855.·52-s + ⋯
L(s)  = 1  − 1.83·2-s + 2.36·4-s + 0.377·7-s − 2.49·8-s + 0.985·11-s + 0.965·13-s − 0.693·14-s + 2.21·16-s − 1.61·17-s + 0.743·19-s − 1.80·22-s + 0.277·23-s − 1.77·26-s + 0.892·28-s − 1.37·29-s + 0.951·31-s − 1.57·32-s + 2.96·34-s − 1.82·37-s − 1.36·38-s + 1.17·41-s − 0.106·43-s + 2.32·44-s − 0.508·46-s − 1.50·47-s + 0.142·49-s + 2.28·52-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 \( 1 + 5.18T + 8T^{2} \)
11 \( 1 - 35.9T + 1.33e3T^{2} \)
13 \( 1 - 45.2T + 2.19e3T^{2} \)
17 \( 1 + 113.T + 4.91e3T^{2} \)
19 \( 1 - 61.5T + 6.85e3T^{2} \)
23 \( 1 - 30.6T + 1.21e4T^{2} \)
29 \( 1 + 214.T + 2.43e4T^{2} \)
31 \( 1 - 164.T + 2.97e4T^{2} \)
37 \( 1 + 410.T + 5.06e4T^{2} \)
41 \( 1 - 309.T + 6.89e4T^{2} \)
43 \( 1 + 29.9T + 7.95e4T^{2} \)
47 \( 1 + 483.T + 1.03e5T^{2} \)
53 \( 1 - 295.T + 1.48e5T^{2} \)
59 \( 1 + 416.T + 2.05e5T^{2} \)
61 \( 1 + 151.T + 2.26e5T^{2} \)
67 \( 1 - 89.5T + 3.00e5T^{2} \)
71 \( 1 + 714.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3T + 3.89e5T^{2} \)
79 \( 1 + 323.T + 4.93e5T^{2} \)
83 \( 1 + 297.T + 5.71e5T^{2} \)
89 \( 1 - 90.2T + 7.04e5T^{2} \)
97 \( 1 + 492.T + 9.12e5T^{2} \)
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

−8.921502478667356202632500361430, −8.101825320904199407378471550997, −7.19921680547288301051593757279, −6.62466394172784678154635173273, −5.78449714296418674283138759688, −4.35555791140189343006475750723, −3.16113950594293333732509025440, −1.90190659764656179828158986516, −1.22691046867859173709849201728, 0, 1.22691046867859173709849201728, 1.90190659764656179828158986516, 3.16113950594293333732509025440, 4.35555791140189343006475750723, 5.78449714296418674283138759688, 6.62466394172784678154635173273, 7.19921680547288301051593757279, 8.101825320904199407378471550997, 8.921502478667356202632500361430

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