Properties

Label 2-3675-1.1-c1-0-17
Degree $2$
Conductor $3675$
Sign $1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93·2-s − 3-s + 1.73·4-s + 1.93·6-s + 0.517·8-s + 9-s − 3.46·11-s − 1.73·12-s + 4·13-s − 4.46·16-s + 4·17-s − 1.93·18-s − 0.378·19-s + 6.69·22-s + 6.31·23-s − 0.517·24-s − 7.72·26-s − 27-s + 8.92·29-s − 7.34·31-s + 7.58·32-s + 3.46·33-s − 7.72·34-s + 1.73·36-s − 0.757·37-s + 0.732·38-s − 4·39-s + ⋯
L(s)  = 1  − 1.36·2-s − 0.577·3-s + 0.866·4-s + 0.788·6-s + 0.183·8-s + 0.333·9-s − 1.04·11-s − 0.500·12-s + 1.10·13-s − 1.11·16-s + 0.970·17-s − 0.455·18-s − 0.0869·19-s + 1.42·22-s + 1.31·23-s − 0.105·24-s − 1.51·26-s − 0.192·27-s + 1.65·29-s − 1.31·31-s + 1.34·32-s + 0.603·33-s − 1.32·34-s + 0.288·36-s − 0.124·37-s + 0.118·38-s − 0.640·39-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6924791331\)
\(L(\frac12)\) \(\approx\) \(0.6924791331\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93T + 2T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 0.378T + 19T^{2} \)
23 \( 1 - 6.31T + 23T^{2} \)
29 \( 1 - 8.92T + 29T^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 + 0.757T + 37T^{2} \)
41 \( 1 + 8.48T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 9.14T + 61T^{2} \)
67 \( 1 + 6.96T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 4.14T + 89T^{2} \)
97 \( 1 - 5.07T + 97T^{2} \)
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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

−8.394945360520378792850589384254, −8.090526636614039865064793708397, −7.12073134325742323519380875362, −6.63345348947785708821396102292, −5.51998244030968874708798296047, −4.99237937445917782148512611504, −3.82553557400490251686701075571, −2.75034245525125024705289118519, −1.51266737127042097706832080440, −0.66289297090853677245546156900, 0.66289297090853677245546156900, 1.51266737127042097706832080440, 2.75034245525125024705289118519, 3.82553557400490251686701075571, 4.99237937445917782148512611504, 5.51998244030968874708798296047, 6.63345348947785708821396102292, 7.12073134325742323519380875362, 8.090526636614039865064793708397, 8.394945360520378792850589384254

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