Properties

Label 2-3675-1.1-c1-0-92
Degree $2$
Conductor $3675$
Sign $-1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
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 + 3-s + 2·4-s − 2·6-s + 9-s − 2·11-s + 2·12-s + 13-s − 4·16-s − 2·18-s − 19-s + 4·22-s − 2·26-s + 27-s + 4·29-s − 9·31-s + 8·32-s − 2·33-s + 2·36-s − 3·37-s + 2·38-s + 39-s + 10·41-s − 5·43-s − 4·44-s − 6·47-s − 4·48-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 0.603·11-s + 0.577·12-s + 0.277·13-s − 16-s − 0.471·18-s − 0.229·19-s + 0.852·22-s − 0.392·26-s + 0.192·27-s + 0.742·29-s − 1.61·31-s + 1.41·32-s − 0.348·33-s + 1/3·36-s − 0.493·37-s + 0.324·38-s + 0.160·39-s + 1.56·41-s − 0.762·43-s − 0.603·44-s − 0.875·47-s − 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(29.3450\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.330290625897708457512030424135, −7.64806652799736373020768972493, −7.07549960992844820900338549539, −6.20891846751957785484478891111, −5.13864962127656346978799713464, −4.23666237765615592023529733657, −3.18274052552709579425202367333, −2.22020796208357033864499560544, −1.34347248663093181873948752914, 0, 1.34347248663093181873948752914, 2.22020796208357033864499560544, 3.18274052552709579425202367333, 4.23666237765615592023529733657, 5.13864962127656346978799713464, 6.20891846751957785484478891111, 7.07549960992844820900338549539, 7.64806652799736373020768972493, 8.330290625897708457512030424135

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