Properties

Label 2-3525-1.1-c1-0-33
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  − 2.24·2-s + 3-s + 3.05·4-s − 2.24·6-s − 1.38·7-s − 2.37·8-s + 9-s + 5.19·11-s + 3.05·12-s + 2.59·13-s + 3.11·14-s − 0.769·16-s − 3.45·17-s − 2.24·18-s + 1.01·19-s − 1.38·21-s − 11.6·22-s − 2.00·23-s − 2.37·24-s − 5.84·26-s + 27-s − 4.23·28-s + 7.23·29-s − 0.286·31-s + 6.48·32-s + 5.19·33-s + 7.76·34-s + ⋯
L(s)  = 1  − 1.59·2-s + 0.577·3-s + 1.52·4-s − 0.918·6-s − 0.523·7-s − 0.840·8-s + 0.333·9-s + 1.56·11-s + 0.882·12-s + 0.720·13-s + 0.832·14-s − 0.192·16-s − 0.837·17-s − 0.530·18-s + 0.233·19-s − 0.302·21-s − 2.48·22-s − 0.417·23-s − 0.485·24-s − 1.14·26-s + 0.192·27-s − 0.800·28-s + 1.34·29-s − 0.0515·31-s + 1.14·32-s + 0.903·33-s + 1.33·34-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: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.129963769\)
\(L(\frac12)\) \(\approx\) \(1.129963769\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.24T + 2T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 - 5.19T + 11T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 - 1.01T + 19T^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 - 7.23T + 29T^{2} \)
31 \( 1 + 0.286T + 31T^{2} \)
37 \( 1 - 4.63T + 37T^{2} \)
41 \( 1 + 0.775T + 41T^{2} \)
43 \( 1 - 2.06T + 43T^{2} \)
53 \( 1 - 0.703T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 7.46T + 61T^{2} \)
67 \( 1 + 3.36T + 67T^{2} \)
71 \( 1 - 3.94T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + 6.69T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.824452793455673823976315787393, −8.116748419202770516978849474767, −7.22052792744990684507961322672, −6.61616669742978548122211607008, −6.08647504714746957092858306147, −4.52243278067205708956638878823, −3.75768154251472985668625463660, −2.71145305996895342378869181861, −1.70038246915362805602152975267, −0.815011824107541595774030110046, 0.815011824107541595774030110046, 1.70038246915362805602152975267, 2.71145305996895342378869181861, 3.75768154251472985668625463660, 4.52243278067205708956638878823, 6.08647504714746957092858306147, 6.61616669742978548122211607008, 7.22052792744990684507961322672, 8.116748419202770516978849474767, 8.824452793455673823976315787393

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