Properties

Label 2-3525-1.1-c1-0-139
Degree $2$
Conductor $3525$
Sign $-1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 3-s + 0.999·4-s + 1.73·6-s + 7-s − 1.73·8-s + 9-s − 4·11-s + 0.999·12-s − 5.73·13-s + 1.73·14-s − 5·16-s − 3.19·17-s + 1.73·18-s + 1.73·19-s + 21-s − 6.92·22-s + 4.46·23-s − 1.73·24-s − 9.92·26-s + 27-s + 0.999·28-s + 0.464·29-s − 3.46·31-s − 5.19·32-s − 4·33-s − 5.53·34-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.707·6-s + 0.377·7-s − 0.612·8-s + 0.333·9-s − 1.20·11-s + 0.288·12-s − 1.58·13-s + 0.462·14-s − 1.25·16-s − 0.775·17-s + 0.408·18-s + 0.397·19-s + 0.218·21-s − 1.47·22-s + 0.930·23-s − 0.353·24-s − 1.94·26-s + 0.192·27-s + 0.188·28-s + 0.0861·29-s − 0.622·31-s − 0.918·32-s − 0.696·33-s − 0.949·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ -1)\)

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 \( 1 \)
47 \( 1 - T \)
good2 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 + 3.19T + 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 - 4.46T + 23T^{2} \)
29 \( 1 - 0.464T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 6.46T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 15.1T + 59T^{2} \)
61 \( 1 + 5.92T + 61T^{2} \)
67 \( 1 - 8.92T + 67T^{2} \)
71 \( 1 - 3.19T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 5.46T + 79T^{2} \)
83 \( 1 - 2.92T + 83T^{2} \)
89 \( 1 + 6.53T + 89T^{2} \)
97 \( 1 - 5.46T + 97T^{2} \)
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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

−8.123872592142635482826933870250, −7.29745520745495085110965811944, −6.78585614540459017253345756543, −5.54943705204085188520559461394, −5.01627813261020809880797425646, −4.54182584650991107472891690243, −3.44135174968241146884130943885, −2.75136717041583464148880957275, −2.00646790922642152845075487646, 0, 2.00646790922642152845075487646, 2.75136717041583464148880957275, 3.44135174968241146884130943885, 4.54182584650991107472891690243, 5.01627813261020809880797425646, 5.54943705204085188520559461394, 6.78585614540459017253345756543, 7.29745520745495085110965811944, 8.123872592142635482826933870250

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