Properties

Label 2-3525-1.1-c1-0-133
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  + 1.60·2-s + 3-s + 0.580·4-s + 1.60·6-s − 2.35·7-s − 2.28·8-s + 9-s + 1.21·11-s + 0.580·12-s + 1.28·13-s − 3.78·14-s − 4.82·16-s − 2.46·17-s + 1.60·18-s − 0.168·19-s − 2.35·21-s + 1.95·22-s − 8.96·23-s − 2.28·24-s + 2.06·26-s + 27-s − 1.36·28-s − 2.39·29-s + 6.32·31-s − 3.18·32-s + 1.21·33-s − 3.95·34-s + ⋯
L(s)  = 1  + 1.13·2-s + 0.577·3-s + 0.290·4-s + 0.655·6-s − 0.889·7-s − 0.806·8-s + 0.333·9-s + 0.367·11-s + 0.167·12-s + 0.357·13-s − 1.01·14-s − 1.20·16-s − 0.596·17-s + 0.378·18-s − 0.0387·19-s − 0.513·21-s + 0.417·22-s − 1.86·23-s − 0.465·24-s + 0.405·26-s + 0.192·27-s − 0.258·28-s − 0.445·29-s + 1.13·31-s − 0.563·32-s + 0.212·33-s − 0.677·34-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: $\chi_{3525} (1, \cdot )$
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 \( 1 - 1.60T + 2T^{2} \)
7 \( 1 + 2.35T + 7T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 - 1.28T + 13T^{2} \)
17 \( 1 + 2.46T + 17T^{2} \)
19 \( 1 + 0.168T + 19T^{2} \)
23 \( 1 + 8.96T + 23T^{2} \)
29 \( 1 + 2.39T + 29T^{2} \)
31 \( 1 - 6.32T + 31T^{2} \)
37 \( 1 + 9.44T + 37T^{2} \)
41 \( 1 + 1.59T + 41T^{2} \)
43 \( 1 + 1.20T + 43T^{2} \)
53 \( 1 + 5.14T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 + 8.76T + 67T^{2} \)
71 \( 1 + 6.83T + 71T^{2} \)
73 \( 1 + 0.476T + 73T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.342178014596554454768243871071, −7.28466564601617962206738789791, −6.37915706689031155743097496565, −6.08281759565483190460909360232, −5.00151911942881307244296190112, −4.17262640216394464255210157905, −3.59868009686670003615862631449, −2.88265102350872203576384543294, −1.84534808731926428958766115825, 0, 1.84534808731926428958766115825, 2.88265102350872203576384543294, 3.59868009686670003615862631449, 4.17262640216394464255210157905, 5.00151911942881307244296190112, 6.08281759565483190460909360232, 6.37915706689031155743097496565, 7.28466564601617962206738789791, 8.342178014596554454768243871071

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