Properties

Label 2-2541-1.1-c1-0-47
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  − 1.56·2-s + 3-s + 0.438·4-s + 5-s − 1.56·6-s + 7-s + 2.43·8-s + 9-s − 1.56·10-s + 0.438·12-s + 5.56·13-s − 1.56·14-s + 15-s − 4.68·16-s + 4.12·17-s − 1.56·18-s + 6·19-s + 0.438·20-s + 21-s − 4·23-s + 2.43·24-s − 4·25-s − 8.68·26-s + 27-s + 0.438·28-s + 6.68·29-s − 1.56·30-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.447·5-s − 0.637·6-s + 0.377·7-s + 0.862·8-s + 0.333·9-s − 0.493·10-s + 0.126·12-s + 1.54·13-s − 0.417·14-s + 0.258·15-s − 1.17·16-s + 0.999·17-s − 0.368·18-s + 1.37·19-s + 0.0980·20-s + 0.218·21-s − 0.834·23-s + 0.497·24-s − 0.800·25-s − 1.70·26-s + 0.192·27-s + 0.0828·28-s + 1.24·29-s − 0.285·30-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

\(L(1)\) \(\approx\) \(1.600869753\)
\(L(\frac12)\) \(\approx\) \(1.600869753\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 2T^{2} \)
5 \( 1 - T + 5T^{2} \)
13 \( 1 - 5.56T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 8.24T + 31T^{2} \)
37 \( 1 + 2.68T + 37T^{2} \)
41 \( 1 + 7.56T + 41T^{2} \)
43 \( 1 + 5.68T + 43T^{2} \)
47 \( 1 - 3.43T + 47T^{2} \)
53 \( 1 - 7.80T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 - 8.80T + 67T^{2} \)
71 \( 1 - 3.12T + 71T^{2} \)
73 \( 1 - 7.12T + 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 + 8.80T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 1.31T + 97T^{2} \)
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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

−8.808791178186706025377966303338, −8.193320392382910427986031826087, −7.83752791360058466545958936186, −6.82315556932055083436163382516, −5.92331233632029478084409522523, −4.98499229255871837576969402770, −3.96545678245895010800091965970, −3.04336360006953761694513366437, −1.70840803898572090112120367954, −1.03934814603548413725743981788, 1.03934814603548413725743981788, 1.70840803898572090112120367954, 3.04336360006953761694513366437, 3.96545678245895010800091965970, 4.98499229255871837576969402770, 5.92331233632029478084409522523, 6.82315556932055083436163382516, 7.83752791360058466545958936186, 8.193320392382910427986031826087, 8.808791178186706025377966303338

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