Properties

Label 2-241-1.1-c1-0-1
Degree $2$
Conductor $241$
Sign $1$
Analytic cond. $1.92439$
Root an. cond. $1.38722$
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  + 0.115·2-s − 3.28·3-s − 1.98·4-s − 1.31·5-s − 0.379·6-s + 3.19·7-s − 0.461·8-s + 7.77·9-s − 0.151·10-s + 1.38·11-s + 6.52·12-s − 5.87·13-s + 0.369·14-s + 4.30·15-s + 3.91·16-s + 5.28·17-s + 0.899·18-s + 4.99·19-s + 2.60·20-s − 10.4·21-s + 0.160·22-s + 3.07·23-s + 1.51·24-s − 3.28·25-s − 0.679·26-s − 15.6·27-s − 6.35·28-s + ⋯
L(s)  = 1  + 0.0817·2-s − 1.89·3-s − 0.993·4-s − 0.586·5-s − 0.155·6-s + 1.20·7-s − 0.163·8-s + 2.59·9-s − 0.0479·10-s + 0.419·11-s + 1.88·12-s − 1.62·13-s + 0.0988·14-s + 1.11·15-s + 0.979·16-s + 1.28·17-s + 0.212·18-s + 1.14·19-s + 0.582·20-s − 2.28·21-s + 0.0342·22-s + 0.640·23-s + 0.309·24-s − 0.656·25-s − 0.133·26-s − 3.01·27-s − 1.20·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(241\)
Sign: $1$
Analytic conductor: \(1.92439\)
Root analytic conductor: \(1.38722\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 241,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5623544471\)
\(L(\frac12)\) \(\approx\) \(0.5623544471\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 - T \)
good2 \( 1 - 0.115T + 2T^{2} \)
3 \( 1 + 3.28T + 3T^{2} \)
5 \( 1 + 1.31T + 5T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + 5.87T + 13T^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
19 \( 1 - 4.99T + 19T^{2} \)
23 \( 1 - 3.07T + 23T^{2} \)
29 \( 1 - 3.28T + 29T^{2} \)
31 \( 1 - 0.672T + 31T^{2} \)
37 \( 1 + 3.79T + 37T^{2} \)
41 \( 1 - 0.970T + 41T^{2} \)
43 \( 1 - 7.93T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 8.45T + 53T^{2} \)
59 \( 1 - 5.70T + 59T^{2} \)
61 \( 1 - 0.717T + 61T^{2} \)
67 \( 1 - 8.81T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + 8.75T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 1.18T + 97T^{2} \)
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

−12.03423662288421035556878010200, −11.45491668813257385004805761554, −10.30604059905496515022833906730, −9.546796007397578905652753495475, −7.909811382413088058571791126211, −7.16374561311484867983482135791, −5.48151733873869996502950363066, −5.06068892700560168139239033355, −4.08777130419848274595431752726, −0.925914816352511531387618261683, 0.925914816352511531387618261683, 4.08777130419848274595431752726, 5.06068892700560168139239033355, 5.48151733873869996502950363066, 7.16374561311484867983482135791, 7.909811382413088058571791126211, 9.546796007397578905652753495475, 10.30604059905496515022833906730, 11.45491668813257385004805761554, 12.03423662288421035556878010200

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