Properties

Label 2-3525-1.1-c1-0-143
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  + 2.04·2-s + 3-s + 2.18·4-s + 2.04·6-s − 1.36·7-s + 0.377·8-s + 9-s − 6.09·11-s + 2.18·12-s + 1.01·13-s − 2.78·14-s − 3.59·16-s − 3.16·17-s + 2.04·18-s − 2.69·19-s − 1.36·21-s − 12.4·22-s − 8.33·23-s + 0.377·24-s + 2.07·26-s + 27-s − 2.97·28-s + 8.92·29-s − 6.61·31-s − 8.11·32-s − 6.09·33-s − 6.46·34-s + ⋯
L(s)  = 1  + 1.44·2-s + 0.577·3-s + 1.09·4-s + 0.835·6-s − 0.515·7-s + 0.133·8-s + 0.333·9-s − 1.83·11-s + 0.630·12-s + 0.281·13-s − 0.745·14-s − 0.899·16-s − 0.766·17-s + 0.482·18-s − 0.617·19-s − 0.297·21-s − 2.66·22-s − 1.73·23-s + 0.0770·24-s + 0.407·26-s + 0.192·27-s − 0.562·28-s + 1.65·29-s − 1.18·31-s − 1.43·32-s − 1.06·33-s − 1.10·34-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 \( 1 - 2.04T + 2T^{2} \)
7 \( 1 + 1.36T + 7T^{2} \)
11 \( 1 + 6.09T + 11T^{2} \)
13 \( 1 - 1.01T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 2.69T + 19T^{2} \)
23 \( 1 + 8.33T + 23T^{2} \)
29 \( 1 - 8.92T + 29T^{2} \)
31 \( 1 + 6.61T + 31T^{2} \)
37 \( 1 - 0.367T + 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 - 8.55T + 43T^{2} \)
53 \( 1 + 0.984T + 53T^{2} \)
59 \( 1 + 1.26T + 59T^{2} \)
61 \( 1 - 3.49T + 61T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 + 6.26T + 73T^{2} \)
79 \( 1 - 8.41T + 79T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−8.129500906828545411450283790969, −7.36184002944074324296440176090, −6.42431191537118808843861484254, −5.88885997703335082899896514453, −5.01194460231373979821023441541, −4.32014334179398973042520035015, −3.56263209411926105574055442682, −2.67765854346050885645588811465, −2.16855245706851192951399306956, 0, 2.16855245706851192951399306956, 2.67765854346050885645588811465, 3.56263209411926105574055442682, 4.32014334179398973042520035015, 5.01194460231373979821023441541, 5.88885997703335082899896514453, 6.42431191537118808843861484254, 7.36184002944074324296440176090, 8.129500906828545411450283790969

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