Properties

Label 2-3525-1.1-c1-0-79
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  − 1.60·2-s + 3-s + 0.590·4-s − 1.60·6-s − 2.89·7-s + 2.26·8-s + 9-s − 2.05·11-s + 0.590·12-s − 4.52·13-s + 4.65·14-s − 4.83·16-s + 4.14·17-s − 1.60·18-s + 3.21·19-s − 2.89·21-s + 3.30·22-s − 2.15·23-s + 2.26·24-s + 7.28·26-s + 27-s − 1.70·28-s + 9.10·29-s + 0.697·31-s + 3.24·32-s − 2.05·33-s − 6.67·34-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.295·4-s − 0.657·6-s − 1.09·7-s + 0.801·8-s + 0.333·9-s − 0.618·11-s + 0.170·12-s − 1.25·13-s + 1.24·14-s − 1.20·16-s + 1.00·17-s − 0.379·18-s + 0.738·19-s − 0.630·21-s + 0.703·22-s − 0.449·23-s + 0.462·24-s + 1.42·26-s + 0.192·27-s − 0.322·28-s + 1.69·29-s + 0.125·31-s + 0.573·32-s − 0.357·33-s − 1.14·34-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 \( 1 + 1.60T + 2T^{2} \)
7 \( 1 + 2.89T + 7T^{2} \)
11 \( 1 + 2.05T + 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 - 3.21T + 19T^{2} \)
23 \( 1 + 2.15T + 23T^{2} \)
29 \( 1 - 9.10T + 29T^{2} \)
31 \( 1 - 0.697T + 31T^{2} \)
37 \( 1 - 8.61T + 37T^{2} \)
41 \( 1 - 3.28T + 41T^{2} \)
43 \( 1 + 4.46T + 43T^{2} \)
53 \( 1 + 2.71T + 53T^{2} \)
59 \( 1 + 8.56T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 + 8.88T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 0.136T + 97T^{2} \)
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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

−8.073074058073846066038322749067, −7.77204685246078191114352881349, −7.05503021309577440775876798774, −6.19030849098906721932332720021, −5.10451740248555701510411930941, −4.32397217096789637347805097833, −3.12020347096287027280422928091, −2.55138374380944435493321749900, −1.20220945587164917344787781257, 0, 1.20220945587164917344787781257, 2.55138374380944435493321749900, 3.12020347096287027280422928091, 4.32397217096789637347805097833, 5.10451740248555701510411930941, 6.19030849098906721932332720021, 7.05503021309577440775876798774, 7.77204685246078191114352881349, 8.073074058073846066038322749067

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