Properties

Label 2-2541-1.1-c1-0-36
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  − 1.46·2-s + 3-s + 0.139·4-s + 2.39·5-s − 1.46·6-s + 7-s + 2.72·8-s + 9-s − 3.50·10-s + 0.139·12-s − 5.04·13-s − 1.46·14-s + 2.39·15-s − 4.25·16-s + 6.36·17-s − 1.46·18-s + 5.32·19-s + 0.333·20-s + 21-s + 4.92·23-s + 2.72·24-s + 0.751·25-s + 7.37·26-s + 27-s + 0.139·28-s − 5.04·29-s − 3.50·30-s + ⋯
L(s)  = 1  − 1.03·2-s + 0.577·3-s + 0.0695·4-s + 1.07·5-s − 0.597·6-s + 0.377·7-s + 0.962·8-s + 0.333·9-s − 1.10·10-s + 0.0401·12-s − 1.39·13-s − 0.390·14-s + 0.619·15-s − 1.06·16-s + 1.54·17-s − 0.344·18-s + 1.22·19-s + 0.0746·20-s + 0.218·21-s + 1.02·23-s + 0.555·24-s + 0.150·25-s + 1.44·26-s + 0.192·27-s + 0.0263·28-s − 0.936·29-s − 0.640·30-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

\(L(1)\) \(\approx\) \(1.612859139\)
\(L(\frac12)\) \(\approx\) \(1.612859139\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.46T + 2T^{2} \)
5 \( 1 - 2.39T + 5T^{2} \)
13 \( 1 + 5.04T + 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 - 4.92T + 23T^{2} \)
29 \( 1 + 5.04T + 29T^{2} \)
31 \( 1 + 7.57T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 0.646T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 0.526T + 47T^{2} \)
53 \( 1 - 3.72T + 53T^{2} \)
59 \( 1 - 7.97T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 1.87T + 97T^{2} \)
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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

−9.244752974055961357669256835345, −8.176302775048576021499208229675, −7.44706189986432594673065513199, −7.15904440373672400643442325469, −5.55266318872004715327565987083, −5.24947098849821418794739458655, −4.04596294837886895479405602491, −2.83742484787363374095014426986, −1.91217864156598817192800680900, −0.976234097004416421234465124479, 0.976234097004416421234465124479, 1.91217864156598817192800680900, 2.83742484787363374095014426986, 4.04596294837886895479405602491, 5.24947098849821418794739458655, 5.55266318872004715327565987083, 7.15904440373672400643442325469, 7.44706189986432594673065513199, 8.176302775048576021499208229675, 9.244752974055961357669256835345

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