Properties

Label 2-2541-1.1-c1-0-28
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  − 0.618·2-s + 3-s − 1.61·4-s + 0.236·5-s − 0.618·6-s − 7-s + 2.23·8-s + 9-s − 0.145·10-s − 1.61·12-s + 1.76·13-s + 0.618·14-s + 0.236·15-s + 1.85·16-s + 4.47·17-s − 0.618·18-s + 3·19-s − 0.381·20-s − 21-s − 0.472·23-s + 2.23·24-s − 4.94·25-s − 1.09·26-s + 27-s + 1.61·28-s − 3·29-s − 0.145·30-s + ⋯
L(s)  = 1  − 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.105·5-s − 0.252·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.0461·10-s − 0.467·12-s + 0.489·13-s + 0.165·14-s + 0.0609·15-s + 0.463·16-s + 1.08·17-s − 0.145·18-s + 0.688·19-s − 0.0854·20-s − 0.218·21-s − 0.0984·23-s + 0.456·24-s − 0.988·25-s − 0.213·26-s + 0.192·27-s + 0.305·28-s − 0.557·29-s − 0.0266·30-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: $\chi_{2541} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451273780\)
\(L(\frac12)\) \(\approx\) \(1.451273780\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
5 \( 1 - 0.236T + 5T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 0.472T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 5.47T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 - 7.47T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 4.70T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 7.76T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.988114701860162842964666795498, −8.091215445843681269208863595386, −7.75286598807876883330113328563, −6.74894279595652241945681546692, −5.69993419467715303090900653717, −5.00709556257650255379608548006, −3.82500538898727930438318084747, −3.39825446398542176220309226827, −1.98827768583556639330875158522, −0.821788043614078500906236176220, 0.821788043614078500906236176220, 1.98827768583556639330875158522, 3.39825446398542176220309226827, 3.82500538898727930438318084747, 5.00709556257650255379608548006, 5.69993419467715303090900653717, 6.74894279595652241945681546692, 7.75286598807876883330113328563, 8.091215445843681269208863595386, 8.988114701860162842964666795498

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