Properties

Label 2-3525-1.1-c1-0-125
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  + 0.237·2-s + 3-s − 1.94·4-s + 0.237·6-s + 1.64·7-s − 0.935·8-s + 9-s + 5.84·11-s − 1.94·12-s − 4.61·13-s + 0.390·14-s + 3.66·16-s − 5.67·17-s + 0.237·18-s − 6.71·19-s + 1.64·21-s + 1.38·22-s − 6.07·23-s − 0.935·24-s − 1.09·26-s + 27-s − 3.20·28-s + 3.23·29-s − 6.81·31-s + 2.74·32-s + 5.84·33-s − 1.34·34-s + ⋯
L(s)  = 1  + 0.167·2-s + 0.577·3-s − 0.971·4-s + 0.0968·6-s + 0.622·7-s − 0.330·8-s + 0.333·9-s + 1.76·11-s − 0.561·12-s − 1.27·13-s + 0.104·14-s + 0.916·16-s − 1.37·17-s + 0.0559·18-s − 1.54·19-s + 0.359·21-s + 0.295·22-s − 1.26·23-s − 0.190·24-s − 0.214·26-s + 0.192·27-s − 0.604·28-s + 0.600·29-s − 1.22·31-s + 0.484·32-s + 1.01·33-s − 0.230·34-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 \( 1 - 0.237T + 2T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 - 5.84T + 11T^{2} \)
13 \( 1 + 4.61T + 13T^{2} \)
17 \( 1 + 5.67T + 17T^{2} \)
19 \( 1 + 6.71T + 19T^{2} \)
23 \( 1 + 6.07T + 23T^{2} \)
29 \( 1 - 3.23T + 29T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 - 5.84T + 37T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 - 0.899T + 43T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 - 6.93T + 59T^{2} \)
61 \( 1 + 6.54T + 61T^{2} \)
67 \( 1 + 9.16T + 67T^{2} \)
71 \( 1 - 3.98T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + 7.03T + 89T^{2} \)
97 \( 1 - 8.59T + 97T^{2} \)
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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

−8.451213944790541615736894862674, −7.57574247885251342445981288016, −6.69887126751503169766607796188, −6.00492986406926032934706096846, −4.80523445142441233786313374908, −4.30128364085845793889453006564, −3.80633321373275284692218020861, −2.45409091950486997231737017383, −1.59687128992922751286071825607, 0, 1.59687128992922751286071825607, 2.45409091950486997231737017383, 3.80633321373275284692218020861, 4.30128364085845793889453006564, 4.80523445142441233786313374908, 6.00492986406926032934706096846, 6.69887126751503169766607796188, 7.57574247885251342445981288016, 8.451213944790541615736894862674

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