Properties

Label 2-3525-1.1-c1-0-127
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  + 0.913·2-s + 3-s − 1.16·4-s + 0.913·6-s + 2.03·7-s − 2.89·8-s + 9-s − 1.09·11-s − 1.16·12-s − 2.41·13-s + 1.86·14-s − 0.308·16-s − 3.31·17-s + 0.913·18-s + 4.97·19-s + 2.03·21-s − 0.997·22-s − 8.10·23-s − 2.89·24-s − 2.20·26-s + 27-s − 2.37·28-s − 7.37·29-s − 3.47·31-s + 5.50·32-s − 1.09·33-s − 3.02·34-s + ⋯
L(s)  = 1  + 0.645·2-s + 0.577·3-s − 0.582·4-s + 0.372·6-s + 0.769·7-s − 1.02·8-s + 0.333·9-s − 0.329·11-s − 0.336·12-s − 0.668·13-s + 0.497·14-s − 0.0771·16-s − 0.804·17-s + 0.215·18-s + 1.14·19-s + 0.444·21-s − 0.212·22-s − 1.68·23-s − 0.590·24-s − 0.431·26-s + 0.192·27-s − 0.448·28-s − 1.36·29-s − 0.623·31-s + 0.972·32-s − 0.190·33-s − 0.519·34-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: Trivial
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 \( 1 - 0.913T + 2T^{2} \)
7 \( 1 - 2.03T + 7T^{2} \)
11 \( 1 + 1.09T + 11T^{2} \)
13 \( 1 + 2.41T + 13T^{2} \)
17 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 + 8.10T + 23T^{2} \)
29 \( 1 + 7.37T + 29T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 + 2.18T + 37T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 - 9.29T + 43T^{2} \)
53 \( 1 + 8.28T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 4.77T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 1.74T + 71T^{2} \)
73 \( 1 + 6.38T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−8.132720655023705532353763148873, −7.64340593762734588764319452779, −6.72418615512735581747999564494, −5.59110132720744212039095253878, −5.16406499814463925017573179604, −4.25887818845322313988537042375, −3.67794230016359596798378482584, −2.64381728669178715824389948245, −1.71416527190471317111478012500, 0, 1.71416527190471317111478012500, 2.64381728669178715824389948245, 3.67794230016359596798378482584, 4.25887818845322313988537042375, 5.16406499814463925017573179604, 5.59110132720744212039095253878, 6.72418615512735581747999564494, 7.64340593762734588764319452779, 8.132720655023705532353763148873

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