Properties

Label 2-3525-1.1-c1-0-80
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  − 0.563·2-s − 3-s − 1.68·4-s + 0.563·6-s + 1.09·7-s + 2.07·8-s + 9-s + 0.827·11-s + 1.68·12-s + 2.66·13-s − 0.617·14-s + 2.19·16-s − 4.16·17-s − 0.563·18-s − 5.48·19-s − 1.09·21-s − 0.466·22-s − 5.61·23-s − 2.07·24-s − 1.50·26-s − 27-s − 1.84·28-s + 9.79·29-s − 9.33·31-s − 5.38·32-s − 0.827·33-s + 2.34·34-s + ⋯
L(s)  = 1  − 0.398·2-s − 0.577·3-s − 0.841·4-s + 0.230·6-s + 0.414·7-s + 0.733·8-s + 0.333·9-s + 0.249·11-s + 0.485·12-s + 0.738·13-s − 0.164·14-s + 0.549·16-s − 1.01·17-s − 0.132·18-s − 1.25·19-s − 0.239·21-s − 0.0993·22-s − 1.17·23-s − 0.423·24-s − 0.294·26-s − 0.192·27-s − 0.348·28-s + 1.81·29-s − 1.67·31-s − 0.952·32-s − 0.144·33-s + 0.402·34-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 \( 1 + 0.563T + 2T^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 - 0.827T + 11T^{2} \)
13 \( 1 - 2.66T + 13T^{2} \)
17 \( 1 + 4.16T + 17T^{2} \)
19 \( 1 + 5.48T + 19T^{2} \)
23 \( 1 + 5.61T + 23T^{2} \)
29 \( 1 - 9.79T + 29T^{2} \)
31 \( 1 + 9.33T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 - 2.92T + 41T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 + 9.17T + 59T^{2} \)
61 \( 1 + 4.17T + 61T^{2} \)
67 \( 1 - 7.79T + 67T^{2} \)
71 \( 1 + 8.28T + 71T^{2} \)
73 \( 1 - 4.51T + 73T^{2} \)
79 \( 1 - 3.96T + 79T^{2} \)
83 \( 1 - 9.53T + 83T^{2} \)
89 \( 1 - 4.86T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−8.294643934225786296775225172325, −7.67000393869569052324212284505, −6.56211315836498809328328035311, −6.06702187646735370933410773819, −5.06233789952297257544039636106, −4.33714052239262040782475743505, −3.84012221021171438264690283833, −2.27242840159609778459365468046, −1.20445986569006262176566468361, 0, 1.20445986569006262176566468361, 2.27242840159609778459365468046, 3.84012221021171438264690283833, 4.33714052239262040782475743505, 5.06233789952297257544039636106, 6.06702187646735370933410773819, 6.56211315836498809328328035311, 7.67000393869569052324212284505, 8.294643934225786296775225172325

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