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Properties

Label 2-45e2-1.1-c1-0-43
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

−8.938465547339551643006764779388, −7.950583101907287210235797238754, −7.53473958441389643062575649292, −6.61917879131764734609394832143, −5.85921211199652148913105101296, −4.56406565613715663118662757462, −3.76761828849360103944041329687, −2.29031487506598717613134720279, −1.40860140644060435443288420559, 0, 1.40860140644060435443288420559, 2.29031487506598717613134720279, 3.76761828849360103944041329687, 4.56406565613715663118662757462, 5.85921211199652148913105101296, 6.61917879131764734609394832143, 7.53473958441389643062575649292, 7.950583101907287210235797238754, 8.938465547339551643006764779388

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