Properties

Label 2-675-1.1-c1-0-15
Degree $2$
Conductor $675$
Sign $-1$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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-s − 4-s + 3·8-s − 5·11-s + 5·13-s − 16-s − 4·17-s − 2·19-s + 5·22-s + 3·23-s − 5·26-s − 10·29-s + 6·31-s − 5·32-s + 4·34-s − 5·37-s + 2·38-s − 10·41-s − 10·43-s + 5·44-s − 3·46-s + 5·47-s − 7·49-s − 5·52-s + 2·53-s + 10·58-s − 5·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.50·11-s + 1.38·13-s − 1/4·16-s − 0.970·17-s − 0.458·19-s + 1.06·22-s + 0.625·23-s − 0.980·26-s − 1.85·29-s + 1.07·31-s − 0.883·32-s + 0.685·34-s − 0.821·37-s + 0.324·38-s − 1.56·41-s − 1.52·43-s + 0.753·44-s − 0.442·46-s + 0.729·47-s − 49-s − 0.693·52-s + 0.274·53-s + 1.31·58-s − 0.650·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -1)\)

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

−10.12687139605402871610685180919, −9.028285709832855333537102502485, −8.470229778473156859055797250625, −7.72941864643843111924093742058, −6.65161085258324315501961849103, −5.45093501870232524860958154077, −4.56767724494133603201732869283, −3.35142569623968027183576325655, −1.76125584480235740498550240381, 0, 1.76125584480235740498550240381, 3.35142569623968027183576325655, 4.56767724494133603201732869283, 5.45093501870232524860958154077, 6.65161085258324315501961849103, 7.72941864643843111924093742058, 8.470229778473156859055797250625, 9.028285709832855333537102502485, 10.12687139605402871610685180919

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