Properties

Label 2-3525-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.25·2-s − 3-s + 3.10·4-s + 2.25·6-s + 3.65·7-s − 2.48·8-s + 9-s − 5.39·11-s − 3.10·12-s + 3.76·13-s − 8.24·14-s − 0.583·16-s − 6.40·17-s − 2.25·18-s − 8.07·19-s − 3.65·21-s + 12.1·22-s − 4.76·23-s + 2.48·24-s − 8.50·26-s − 27-s + 11.3·28-s − 3.78·29-s − 5.26·31-s + 6.29·32-s + 5.39·33-s + 14.4·34-s + ⋯
L(s)  = 1  − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.922·6-s + 1.38·7-s − 0.879·8-s + 0.333·9-s − 1.62·11-s − 0.895·12-s + 1.04·13-s − 2.20·14-s − 0.145·16-s − 1.55·17-s − 0.532·18-s − 1.85·19-s − 0.797·21-s + 2.59·22-s − 0.993·23-s + 0.507·24-s − 1.66·26-s − 0.192·27-s + 2.14·28-s − 0.702·29-s − 0.944·31-s + 1.11·32-s + 0.938·33-s + 2.48·34-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: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4565448840\)
\(L(\frac12)\) \(\approx\) \(0.4565448840\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.25T + 2T^{2} \)
7 \( 1 - 3.65T + 7T^{2} \)
11 \( 1 + 5.39T + 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 + 8.07T + 19T^{2} \)
23 \( 1 + 4.76T + 23T^{2} \)
29 \( 1 + 3.78T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 - 3.48T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 5.28T + 43T^{2} \)
53 \( 1 + 9.62T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 4.68T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 1.64T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 7.50T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.493775410446760632947405609122, −8.011119521715102470383174390148, −7.44876091578675731734212005981, −6.48511976285172257360578883769, −5.79641226223031473750999038170, −4.77430376312352453949168613063, −4.10559786328867350870880364211, −2.23380511862968225243704453166, −1.93091815465900330252277677117, −0.50549684789553836895072413164, 0.50549684789553836895072413164, 1.93091815465900330252277677117, 2.23380511862968225243704453166, 4.10559786328867350870880364211, 4.77430376312352453949168613063, 5.79641226223031473750999038170, 6.48511976285172257360578883769, 7.44876091578675731734212005981, 8.011119521715102470383174390148, 8.493775410446760632947405609122

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