Properties

Label 2-45e2-1.1-c3-0-220
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  + 4.85·2-s + 15.5·4-s + 16.2·7-s + 36.5·8-s − 59.5·11-s + 27.0·13-s + 78.7·14-s + 53.0·16-s − 78.9·17-s − 142.·19-s − 288.·22-s − 85.6·23-s + 131.·26-s + 252.·28-s − 226.·29-s − 150.·31-s − 35.0·32-s − 382.·34-s + 234.·37-s − 690.·38-s − 73.8·41-s + 267.·43-s − 924.·44-s − 415.·46-s − 266.·47-s − 79.3·49-s + 420.·52-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.94·4-s + 0.876·7-s + 1.61·8-s − 1.63·11-s + 0.577·13-s + 1.50·14-s + 0.829·16-s − 1.12·17-s − 1.71·19-s − 2.79·22-s − 0.776·23-s + 0.990·26-s + 1.70·28-s − 1.44·29-s − 0.873·31-s − 0.193·32-s − 1.93·34-s + 1.04·37-s − 2.94·38-s − 0.281·41-s + 0.947·43-s − 3.16·44-s − 1.33·46-s − 0.825·47-s − 0.231·49-s + 1.12·52-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 \( 1 - 4.85T + 8T^{2} \)
7 \( 1 - 16.2T + 343T^{2} \)
11 \( 1 + 59.5T + 1.33e3T^{2} \)
13 \( 1 - 27.0T + 2.19e3T^{2} \)
17 \( 1 + 78.9T + 4.91e3T^{2} \)
19 \( 1 + 142.T + 6.85e3T^{2} \)
23 \( 1 + 85.6T + 1.21e4T^{2} \)
29 \( 1 + 226.T + 2.43e4T^{2} \)
31 \( 1 + 150.T + 2.97e4T^{2} \)
37 \( 1 - 234.T + 5.06e4T^{2} \)
41 \( 1 + 73.8T + 6.89e4T^{2} \)
43 \( 1 - 267.T + 7.95e4T^{2} \)
47 \( 1 + 266.T + 1.03e5T^{2} \)
53 \( 1 - 603.T + 1.48e5T^{2} \)
59 \( 1 + 508.T + 2.05e5T^{2} \)
61 \( 1 - 78.3T + 2.26e5T^{2} \)
67 \( 1 - 488.T + 3.00e5T^{2} \)
71 \( 1 - 73.2T + 3.57e5T^{2} \)
73 \( 1 - 115.T + 3.89e5T^{2} \)
79 \( 1 - 782.T + 4.93e5T^{2} \)
83 \( 1 - 271.T + 5.71e5T^{2} \)
89 \( 1 + 211.T + 7.04e5T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−8.141843490324596630511408488858, −7.49815729032289622690229993449, −6.47972658223186164116026086338, −5.80361300755507250787413278101, −5.06523750313695219531025110687, −4.37726297754965478785202165392, −3.67455174563280715953167162220, −2.38657425812594927714958365594, −1.99088967485610600731475839076, 0, 1.99088967485610600731475839076, 2.38657425812594927714958365594, 3.67455174563280715953167162220, 4.37726297754965478785202165392, 5.06523750313695219531025110687, 5.80361300755507250787413278101, 6.47972658223186164116026086338, 7.49815729032289622690229993449, 8.141843490324596630511408488858

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