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Properties

Label 2-45e2-1.1-c1-0-46
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  − 2·4-s − 2·7-s + 3·11-s + 4·13-s + 4·16-s − 6·17-s − 19-s − 6·23-s + 4·28-s + 9·29-s − 31-s − 8·37-s − 3·41-s + 4·43-s − 6·44-s + 12·47-s − 3·49-s − 8·52-s + 6·53-s − 3·59-s − 10·61-s − 8·64-s − 14·67-s + 12·68-s + 3·71-s − 2·73-s + 2·76-s + ⋯
L(s)  = 1  − 4-s − 0.755·7-s + 0.904·11-s + 1.10·13-s + 16-s − 1.45·17-s − 0.229·19-s − 1.25·23-s + 0.755·28-s + 1.67·29-s − 0.179·31-s − 1.31·37-s − 0.468·41-s + 0.609·43-s − 0.904·44-s + 1.75·47-s − 3/7·49-s − 1.10·52-s + 0.824·53-s − 0.390·59-s − 1.28·61-s − 64-s − 1.71·67-s + 1.45·68-s + 0.356·71-s − 0.234·73-s + 0.229·76-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 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.767747284753786098117223574742, −8.335548836313514290240539244878, −7.05773573312300104815129757004, −6.32468375473848937056457133649, −5.68073732624305661536135613556, −4.35117009626133411095198923203, −4.03163071651258240001742541940, −2.94203711656515483067854528639, −1.43817397608101885315913429842, 0, 1.43817397608101885315913429842, 2.94203711656515483067854528639, 4.03163071651258240001742541940, 4.35117009626133411095198923203, 5.68073732624305661536135613556, 6.32468375473848937056457133649, 7.05773573312300104815129757004, 8.335548836313514290240539244878, 8.767747284753786098117223574742

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