Properties

Label 2-3525-1.1-c1-0-83
Degree $2$
Conductor $3525$
Sign $1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 2·6-s + 7-s + 9-s + 4·11-s + 2·12-s − 13-s + 2·14-s − 4·16-s + 4·17-s + 2·18-s + 5·19-s + 21-s + 8·22-s + 2·23-s − 2·26-s + 27-s + 2·28-s − 2·29-s − 5·31-s − 8·32-s + 4·33-s + 8·34-s + 2·36-s + 6·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 0.277·13-s + 0.534·14-s − 16-s + 0.970·17-s + 0.471·18-s + 1.14·19-s + 0.218·21-s + 1.70·22-s + 0.417·23-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 0.898·31-s − 1.41·32-s + 0.696·33-s + 1.37·34-s + 1/3·36-s + 0.986·37-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.672940052\)
\(L(\frac12)\) \(\approx\) \(5.672940052\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−8.531034119231606997358379947889, −7.64159255651979498527664543507, −6.98329947880719090673663692701, −6.17572641567605170291097468578, −5.34551916446366232334016868312, −4.75683828410938314398500928825, −3.75526031471002211951925005045, −3.40064887215716970134281108990, −2.35242390355512730670024614548, −1.23151358695066419893154741695, 1.23151358695066419893154741695, 2.35242390355512730670024614548, 3.40064887215716970134281108990, 3.75526031471002211951925005045, 4.75683828410938314398500928825, 5.34551916446366232334016868312, 6.17572641567605170291097468578, 6.98329947880719090673663692701, 7.64159255651979498527664543507, 8.531034119231606997358379947889

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