Properties

Label 2-45e2-1.1-c3-0-93
Degree $2$
Conductor $2025$
Sign $-1$
Analytic cond. $119.478$
Root an. cond. $10.9306$
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  − 5.54·2-s + 22.7·4-s − 25.7·7-s − 81.7·8-s − 6.25·11-s + 15.2·13-s + 142.·14-s + 271.·16-s − 36.0·17-s − 52.7·19-s + 34.6·22-s − 83.7·23-s − 84.6·26-s − 584.·28-s − 119.·29-s + 276.·31-s − 849.·32-s + 199.·34-s + 117.·37-s + 292.·38-s + 159.·41-s + 294.·43-s − 142.·44-s + 464.·46-s + 83.2·47-s + 318.·49-s + 346.·52-s + ⋯
L(s)  = 1  − 1.96·2-s + 2.84·4-s − 1.38·7-s − 3.61·8-s − 0.171·11-s + 0.325·13-s + 2.72·14-s + 4.23·16-s − 0.513·17-s − 0.636·19-s + 0.336·22-s − 0.759·23-s − 0.638·26-s − 3.94·28-s − 0.762·29-s + 1.60·31-s − 4.69·32-s + 1.00·34-s + 0.522·37-s + 1.24·38-s + 0.606·41-s + 1.04·43-s − 0.487·44-s + 1.48·46-s + 0.258·47-s + 0.927·49-s + 0.925·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(119.478\)
Root analytic conductor: \(10.9306\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2025,\ (\ :3/2),\ -1)\)

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 \( 1 \)
good2 \( 1 + 5.54T + 8T^{2} \)
7 \( 1 + 25.7T + 343T^{2} \)
11 \( 1 + 6.25T + 1.33e3T^{2} \)
13 \( 1 - 15.2T + 2.19e3T^{2} \)
17 \( 1 + 36.0T + 4.91e3T^{2} \)
19 \( 1 + 52.7T + 6.85e3T^{2} \)
23 \( 1 + 83.7T + 1.21e4T^{2} \)
29 \( 1 + 119.T + 2.43e4T^{2} \)
31 \( 1 - 276.T + 2.97e4T^{2} \)
37 \( 1 - 117.T + 5.06e4T^{2} \)
41 \( 1 - 159.T + 6.89e4T^{2} \)
43 \( 1 - 294.T + 7.95e4T^{2} \)
47 \( 1 - 83.2T + 1.03e5T^{2} \)
53 \( 1 - 149.T + 1.48e5T^{2} \)
59 \( 1 + 635.T + 2.05e5T^{2} \)
61 \( 1 - 597.T + 2.26e5T^{2} \)
67 \( 1 - 330.T + 3.00e5T^{2} \)
71 \( 1 - 1.14e3T + 3.57e5T^{2} \)
73 \( 1 - 130.T + 3.89e5T^{2} \)
79 \( 1 + 737.T + 4.93e5T^{2} \)
83 \( 1 + 369.T + 5.71e5T^{2} \)
89 \( 1 - 225.T + 7.04e5T^{2} \)
97 \( 1 - 11.9T + 9.12e5T^{2} \)
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

−8.467140050375508582488172547343, −7.86933339589324230169405878093, −6.93688317749217786667623981126, −6.38966679242229755798199360988, −5.78287294147640288613307947410, −3.97042652740245127562257509866, −2.87629457116545070891221000050, −2.17704435170297860396787945654, −0.864771407927055956235531452416, 0, 0.864771407927055956235531452416, 2.17704435170297860396787945654, 2.87629457116545070891221000050, 3.97042652740245127562257509866, 5.78287294147640288613307947410, 6.38966679242229755798199360988, 6.93688317749217786667623981126, 7.86933339589324230169405878093, 8.467140050375508582488172547343

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