Properties

Label 2-2541-1.1-c1-0-27
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s + 3-s + 0.618·4-s − 4.23·5-s + 1.61·6-s − 7-s − 2.23·8-s + 9-s − 6.85·10-s + 0.618·12-s + 6.23·13-s − 1.61·14-s − 4.23·15-s − 4.85·16-s − 4.47·17-s + 1.61·18-s + 3·19-s − 2.61·20-s − 21-s + 8.47·23-s − 2.23·24-s + 12.9·25-s + 10.0·26-s + 27-s − 0.618·28-s − 3·29-s − 6.85·30-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.577·3-s + 0.309·4-s − 1.89·5-s + 0.660·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s − 2.16·10-s + 0.178·12-s + 1.72·13-s − 0.432·14-s − 1.09·15-s − 1.21·16-s − 1.08·17-s + 0.381·18-s + 0.688·19-s − 0.585·20-s − 0.218·21-s + 1.76·23-s − 0.456·24-s + 2.58·25-s + 1.97·26-s + 0.192·27-s − 0.116·28-s − 0.557·29-s − 1.25·30-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

\(L(1)\) \(\approx\) \(2.433769256\)
\(L(\frac12)\) \(\approx\) \(2.433769256\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 2T^{2} \)
5 \( 1 + 4.23T + 5T^{2} \)
13 \( 1 - 6.23T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 3.47T + 37T^{2} \)
41 \( 1 - 1.52T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 - 8.94T + 53T^{2} \)
59 \( 1 + 1.47T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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

−8.857143704386704643850720887222, −8.170794739156512386351255390664, −7.24857412388667026438192650925, −6.66498985060980633707361761410, −5.64248568446366900615917976346, −4.58447984416195775358413899347, −4.02555076456381273504056017435, −3.43870567717536772254314128943, −2.79248104188882076909598089751, −0.814502883612581387160443608813, 0.814502883612581387160443608813, 2.79248104188882076909598089751, 3.43870567717536772254314128943, 4.02555076456381273504056017435, 4.58447984416195775358413899347, 5.64248568446366900615917976346, 6.66498985060980633707361761410, 7.24857412388667026438192650925, 8.170794739156512386351255390664, 8.857143704386704643850720887222

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