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Properties

Label 2-45e2-1.1-c1-0-68
Degree $2$
Conductor $2025$
Sign $-1$
Analytic cond. $16.1697$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 5·11-s − 4·13-s − 4·16-s − 4·17-s − 5·19-s − 10·22-s + 6·23-s − 8·26-s + 5·29-s − 9·31-s − 8·32-s − 8·34-s + 10·37-s − 10·38-s − 7·41-s + 2·43-s − 10·44-s + 12·46-s + 2·47-s − 7·49-s − 8·52-s + 8·53-s + 10·58-s + 59-s − 2·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.50·11-s − 1.10·13-s − 16-s − 0.970·17-s − 1.14·19-s − 2.13·22-s + 1.25·23-s − 1.56·26-s + 0.928·29-s − 1.61·31-s − 1.41·32-s − 1.37·34-s + 1.64·37-s − 1.62·38-s − 1.09·41-s + 0.304·43-s − 1.50·44-s + 1.76·46-s + 0.291·47-s − 49-s − 1.10·52-s + 1.09·53-s + 1.31·58-s + 0.130·59-s − 0.256·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(16.1697\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.739862108778831116392306497161, −7.78161059086003591989398023564, −6.95020224721867641073570394495, −6.23779450852815458911834047586, −5.16339567293851105751777832527, −4.88501239146098945544230145462, −3.94615232718651422948486243217, −2.78226659598101469504401585183, −2.28955250042389927046640802946, 0, 2.28955250042389927046640802946, 2.78226659598101469504401585183, 3.94615232718651422948486243217, 4.88501239146098945544230145462, 5.16339567293851105751777832527, 6.23779450852815458911834047586, 6.95020224721867641073570394495, 7.78161059086003591989398023564, 8.739862108778831116392306497161

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