Properties

Label 2-2541-1.1-c1-0-53
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  + 0.470·2-s + 3-s − 1.77·4-s + 3.24·5-s + 0.470·6-s + 7-s − 1.77·8-s + 9-s + 1.52·10-s − 1.77·12-s + 2.47·13-s + 0.470·14-s + 3.24·15-s + 2.71·16-s + 3.24·17-s + 0.470·18-s + 5.30·19-s − 5.77·20-s + 21-s − 8.86·23-s − 1.77·24-s + 5.55·25-s + 1.16·26-s + 27-s − 1.77·28-s + 2.47·29-s + 1.52·30-s + ⋯
L(s)  = 1  + 0.332·2-s + 0.577·3-s − 0.889·4-s + 1.45·5-s + 0.192·6-s + 0.377·7-s − 0.628·8-s + 0.333·9-s + 0.483·10-s − 0.513·12-s + 0.685·13-s + 0.125·14-s + 0.838·15-s + 0.679·16-s + 0.788·17-s + 0.110·18-s + 1.21·19-s − 1.29·20-s + 0.218·21-s − 1.84·23-s − 0.363·24-s + 1.11·25-s + 0.228·26-s + 0.192·27-s − 0.336·28-s + 0.458·29-s + 0.279·30-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

\(L(1)\) \(\approx\) \(3.114597524\)
\(L(\frac12)\) \(\approx\) \(3.114597524\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.470T + 2T^{2} \)
5 \( 1 - 3.24T + 5T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
17 \( 1 - 3.24T + 17T^{2} \)
19 \( 1 - 5.30T + 19T^{2} \)
23 \( 1 + 8.86T + 23T^{2} \)
29 \( 1 - 2.47T + 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 - 1.28T + 41T^{2} \)
43 \( 1 - 4.89T + 43T^{2} \)
47 \( 1 - 3.96T + 47T^{2} \)
53 \( 1 + 9.77T + 53T^{2} \)
59 \( 1 - 2.41T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 3.19T + 71T^{2} \)
73 \( 1 + 8.98T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 7.86T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−9.079810782772306433040007366831, −8.191388093932423944306159311034, −7.59500315597590810745959176149, −6.28333013505408432340829300036, −5.70849183972623060521825232060, −5.09073091798894629558670744057, −4.03761720098368662481946467660, −3.26294629659885011832102448882, −2.14012035867834843536199380847, −1.14612272401292708000285316324, 1.14612272401292708000285316324, 2.14012035867834843536199380847, 3.26294629659885011832102448882, 4.03761720098368662481946467660, 5.09073091798894629558670744057, 5.70849183972623060521825232060, 6.28333013505408432340829300036, 7.59500315597590810745959176149, 8.191388093932423944306159311034, 9.079810782772306433040007366831

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