Properties

Label 2-2541-1.1-c1-0-62
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.301·2-s − 3-s − 1.90·4-s + 1.30·5-s + 0.301·6-s − 7-s + 1.17·8-s + 9-s − 0.391·10-s + 1.90·12-s − 3.38·13-s + 0.301·14-s − 1.30·15-s + 3.46·16-s + 0.478·17-s − 0.301·18-s + 1.60·19-s − 2.48·20-s + 21-s + 3.47·23-s − 1.17·24-s − 3.30·25-s + 1.02·26-s − 27-s + 1.90·28-s + 0.942·29-s + 0.391·30-s + ⋯
L(s)  = 1  − 0.212·2-s − 0.577·3-s − 0.954·4-s + 0.581·5-s + 0.122·6-s − 0.377·7-s + 0.416·8-s + 0.333·9-s − 0.123·10-s + 0.551·12-s − 0.939·13-s + 0.0804·14-s − 0.335·15-s + 0.866·16-s + 0.116·17-s − 0.0709·18-s + 0.368·19-s − 0.555·20-s + 0.218·21-s + 0.725·23-s − 0.240·24-s − 0.661·25-s + 0.200·26-s − 0.192·27-s + 0.360·28-s + 0.175·29-s + 0.0715·30-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: \(1\)
Selberg data: \((2,\ 2541,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.301T + 2T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
13 \( 1 + 3.38T + 13T^{2} \)
17 \( 1 - 0.478T + 17T^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 - 0.942T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + 3.55T + 37T^{2} \)
41 \( 1 - 3.94T + 41T^{2} \)
43 \( 1 + 2.87T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 5.28T + 53T^{2} \)
59 \( 1 + 9.69T + 59T^{2} \)
61 \( 1 - 3.00T + 61T^{2} \)
67 \( 1 - 7.74T + 67T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + 8.86T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 7.54T + 89T^{2} \)
97 \( 1 + 1.71T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.777976887041693018761631101296, −7.65284262463712697273740052492, −7.10093287500792470053624327876, −5.99988244563267612562999385313, −5.41288975483105111454723797133, −4.68110678475560861832658488912, −3.79560237769095641987918470962, −2.61978102009251555080902786297, −1.27636926070606124792581087976, 0, 1.27636926070606124792581087976, 2.61978102009251555080902786297, 3.79560237769095641987918470962, 4.68110678475560861832658488912, 5.41288975483105111454723797133, 5.99988244563267612562999385313, 7.10093287500792470053624327876, 7.65284262463712697273740052492, 8.777976887041693018761631101296

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