Properties

Label 2-45e2-1.1-c3-0-137
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.57·2-s + 12.9·4-s + 20.1·7-s − 22.4·8-s − 66.3·11-s + 46.8·13-s − 91.9·14-s − 0.475·16-s − 47.6·17-s − 9.95·19-s + 303.·22-s + 9.59·23-s − 214.·26-s + 259.·28-s + 178.·29-s + 154.·31-s + 182.·32-s + 217.·34-s − 248.·37-s + 45.5·38-s − 249.·41-s + 212.·43-s − 856.·44-s − 43.8·46-s + 475.·47-s + 61.5·49-s + 604.·52-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.61·4-s + 1.08·7-s − 0.993·8-s − 1.81·11-s + 0.998·13-s − 1.75·14-s − 0.00743·16-s − 0.679·17-s − 0.120·19-s + 2.94·22-s + 0.0869·23-s − 1.61·26-s + 1.75·28-s + 1.14·29-s + 0.892·31-s + 1.00·32-s + 1.09·34-s − 1.10·37-s + 0.194·38-s − 0.951·41-s + 0.752·43-s − 2.93·44-s − 0.140·46-s + 1.47·47-s + 0.179·49-s + 1.61·52-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 \( 1 + 4.57T + 8T^{2} \)
7 \( 1 - 20.1T + 343T^{2} \)
11 \( 1 + 66.3T + 1.33e3T^{2} \)
13 \( 1 - 46.8T + 2.19e3T^{2} \)
17 \( 1 + 47.6T + 4.91e3T^{2} \)
19 \( 1 + 9.95T + 6.85e3T^{2} \)
23 \( 1 - 9.59T + 1.21e4T^{2} \)
29 \( 1 - 178.T + 2.43e4T^{2} \)
31 \( 1 - 154.T + 2.97e4T^{2} \)
37 \( 1 + 248.T + 5.06e4T^{2} \)
41 \( 1 + 249.T + 6.89e4T^{2} \)
43 \( 1 - 212.T + 7.95e4T^{2} \)
47 \( 1 - 475.T + 1.03e5T^{2} \)
53 \( 1 + 546.T + 1.48e5T^{2} \)
59 \( 1 - 419.T + 2.05e5T^{2} \)
61 \( 1 + 545.T + 2.26e5T^{2} \)
67 \( 1 - 447.T + 3.00e5T^{2} \)
71 \( 1 + 409.T + 3.57e5T^{2} \)
73 \( 1 - 358.T + 3.89e5T^{2} \)
79 \( 1 + 651.T + 4.93e5T^{2} \)
83 \( 1 + 813.T + 5.71e5T^{2} \)
89 \( 1 - 200.T + 7.04e5T^{2} \)
97 \( 1 + 252.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.404407934498374254359017448872, −7.955582010399594690790616968643, −7.20284919956292269188844262206, −6.29591152354958263748520176348, −5.23181864242403240813993775295, −4.43551558525317428885985577200, −2.89090847501305120018921078315, −2.03439014167654350859011633740, −1.08589461222861033705591123322, 0, 1.08589461222861033705591123322, 2.03439014167654350859011633740, 2.89090847501305120018921078315, 4.43551558525317428885985577200, 5.23181864242403240813993775295, 6.29591152354958263748520176348, 7.20284919956292269188844262206, 7.955582010399594690790616968643, 8.404407934498374254359017448872

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