Properties

Label 2-3525-1.1-c1-0-131
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  + 1.68·2-s + 3-s + 0.841·4-s + 1.68·6-s − 4.79·7-s − 1.95·8-s + 9-s + 4.98·11-s + 0.841·12-s + 3.91·13-s − 8.08·14-s − 4.97·16-s − 6.51·17-s + 1.68·18-s − 4.65·19-s − 4.79·21-s + 8.40·22-s + 1.62·23-s − 1.95·24-s + 6.59·26-s + 27-s − 4.03·28-s + 3.76·29-s − 7.29·31-s − 4.47·32-s + 4.98·33-s − 10.9·34-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.577·3-s + 0.420·4-s + 0.688·6-s − 1.81·7-s − 0.690·8-s + 0.333·9-s + 1.50·11-s + 0.242·12-s + 1.08·13-s − 2.16·14-s − 1.24·16-s − 1.57·17-s + 0.397·18-s − 1.06·19-s − 1.04·21-s + 1.79·22-s + 0.339·23-s − 0.398·24-s + 1.29·26-s + 0.192·27-s − 0.762·28-s + 0.699·29-s − 1.31·31-s − 0.791·32-s + 0.868·33-s − 1.88·34-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 \( 1 - 1.68T + 2T^{2} \)
7 \( 1 + 4.79T + 7T^{2} \)
11 \( 1 - 4.98T + 11T^{2} \)
13 \( 1 - 3.91T + 13T^{2} \)
17 \( 1 + 6.51T + 17T^{2} \)
19 \( 1 + 4.65T + 19T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 - 3.76T + 29T^{2} \)
31 \( 1 + 7.29T + 31T^{2} \)
37 \( 1 + 8.07T + 37T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 + 2.49T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 - 6.18T + 67T^{2} \)
71 \( 1 - 9.54T + 71T^{2} \)
73 \( 1 - 4.36T + 73T^{2} \)
79 \( 1 - 3.21T + 79T^{2} \)
83 \( 1 + 5.71T + 83T^{2} \)
89 \( 1 + 1.60T + 89T^{2} \)
97 \( 1 - 8.26T + 97T^{2} \)
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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

−8.537305809126125293963175107499, −6.88609774868796798910628427357, −6.55028640750459644382139487666, −6.20723504525777995257010444854, −5.00221149780268878701081449350, −3.97834583410543727237989832286, −3.68920749053424735481698602283, −2.97430301927446834008143164450, −1.82245990115195922441416620902, 0, 1.82245990115195922441416620902, 2.97430301927446834008143164450, 3.68920749053424735481698602283, 3.97834583410543727237989832286, 5.00221149780268878701081449350, 6.20723504525777995257010444854, 6.55028640750459644382139487666, 6.88609774868796798910628427357, 8.537305809126125293963175107499

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