Properties

Label 2-2541-1.1-c1-0-43
Degree $2$
Conductor $2541$
Sign $1$
Analytic cond. $20.2899$
Root an. cond. $4.50444$
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  + 1.32·2-s + 3-s − 0.239·4-s + 0.326·5-s + 1.32·6-s − 7-s − 2.97·8-s + 9-s + 0.433·10-s − 0.239·12-s − 0.0589·13-s − 1.32·14-s + 0.326·15-s − 3.46·16-s + 6.28·17-s + 1.32·18-s + 2.43·19-s − 0.0782·20-s − 21-s + 5.93·23-s − 2.97·24-s − 4.89·25-s − 0.0782·26-s + 27-s + 0.239·28-s + 9.43·29-s + 0.433·30-s + ⋯
L(s)  = 1  + 0.938·2-s + 0.577·3-s − 0.119·4-s + 0.146·5-s + 0.541·6-s − 0.377·7-s − 1.05·8-s + 0.333·9-s + 0.137·10-s − 0.0690·12-s − 0.0163·13-s − 0.354·14-s + 0.0844·15-s − 0.866·16-s + 1.52·17-s + 0.312·18-s + 0.558·19-s − 0.0174·20-s − 0.218·21-s + 1.23·23-s − 0.606·24-s − 0.978·25-s − 0.0153·26-s + 0.192·27-s + 0.0452·28-s + 1.75·29-s + 0.0791·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(20.2899\)
Root analytic conductor: \(4.50444\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.225990761\)
\(L(\frac12)\) \(\approx\) \(3.225990761\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 1.32T + 2T^{2} \)
5 \( 1 - 0.326T + 5T^{2} \)
13 \( 1 + 0.0589T + 13T^{2} \)
17 \( 1 - 6.28T + 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
23 \( 1 - 5.93T + 23T^{2} \)
29 \( 1 - 9.43T + 29T^{2} \)
31 \( 1 - 1.98T + 31T^{2} \)
37 \( 1 + 6.53T + 37T^{2} \)
41 \( 1 + 0.0782T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 - 6.01T + 47T^{2} \)
53 \( 1 + 5.94T + 53T^{2} \)
59 \( 1 - 1.13T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 8.75T + 67T^{2} \)
71 \( 1 + 1.65T + 71T^{2} \)
73 \( 1 - 9.40T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 7.49T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 6.71T + 97T^{2} \)
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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

−8.873098696257806529625207598281, −8.210693071958330554767265892506, −7.29472318339927871129214899884, −6.47410961223211854268125980772, −5.57857385569260262819008063124, −4.99305096851432735166275557852, −3.98429375328613011950614477817, −3.27696248668565379865974303419, −2.60073375527448046980541230878, −1.00541820926269119317767085626, 1.00541820926269119317767085626, 2.60073375527448046980541230878, 3.27696248668565379865974303419, 3.98429375328613011950614477817, 4.99305096851432735166275557852, 5.57857385569260262819008063124, 6.47410961223211854268125980772, 7.29472318339927871129214899884, 8.210693071958330554767265892506, 8.873098696257806529625207598281

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