Properties

Label 2-3525-1.1-c1-0-120
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  − 0.936·2-s + 3-s − 1.12·4-s − 0.936·6-s + 3.65·7-s + 2.92·8-s + 9-s − 5.57·11-s − 1.12·12-s + 2.92·13-s − 3.42·14-s − 0.494·16-s + 2.26·17-s − 0.936·18-s − 2.64·19-s + 3.65·21-s + 5.22·22-s − 7.72·23-s + 2.92·24-s − 2.74·26-s + 27-s − 4.10·28-s − 7.79·29-s − 6.03·31-s − 5.38·32-s − 5.57·33-s − 2.12·34-s + ⋯
L(s)  = 1  − 0.662·2-s + 0.577·3-s − 0.561·4-s − 0.382·6-s + 1.38·7-s + 1.03·8-s + 0.333·9-s − 1.68·11-s − 0.324·12-s + 0.812·13-s − 0.915·14-s − 0.123·16-s + 0.549·17-s − 0.220·18-s − 0.606·19-s + 0.797·21-s + 1.11·22-s − 1.61·23-s + 0.597·24-s − 0.538·26-s + 0.192·27-s − 0.775·28-s − 1.44·29-s − 1.08·31-s − 0.952·32-s − 0.970·33-s − 0.363·34-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 \( 1 + 0.936T + 2T^{2} \)
7 \( 1 - 3.65T + 7T^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 - 2.26T + 17T^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 + 7.72T + 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 + 6.03T + 31T^{2} \)
37 \( 1 - 6.60T + 37T^{2} \)
41 \( 1 + 8.81T + 41T^{2} \)
43 \( 1 + 9.69T + 43T^{2} \)
53 \( 1 - 2.97T + 53T^{2} \)
59 \( 1 - 0.571T + 59T^{2} \)
61 \( 1 - 8.30T + 61T^{2} \)
67 \( 1 - 8.03T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 5.94T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
97 \( 1 + 5.24T + 97T^{2} \)
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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

−8.383329242826412550638141720245, −7.82597246658139787041713353420, −7.16280365468827635213386556512, −5.70023351507480963278515526896, −5.19400905714452876079943176586, −4.30062776318467273911310778081, −3.56554348111583456114898137555, −2.18580449455154231732463787712, −1.53095065260380383555879054465, 0, 1.53095065260380383555879054465, 2.18580449455154231732463787712, 3.56554348111583456114898137555, 4.30062776318467273911310778081, 5.19400905714452876079943176586, 5.70023351507480963278515526896, 7.16280365468827635213386556512, 7.82597246658139787041713353420, 8.383329242826412550638141720245

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