Properties

Label 2-1881-1.1-c1-0-12
Degree $2$
Conductor $1881$
Sign $1$
Analytic cond. $15.0198$
Root an. cond. $3.87554$
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  − 1.82·2-s + 1.33·4-s − 2.34·5-s + 1.69·7-s + 1.21·8-s + 4.28·10-s − 11-s + 4.11·13-s − 3.09·14-s − 4.88·16-s + 6.16·17-s − 19-s − 3.13·20-s + 1.82·22-s − 3.52·23-s + 0.502·25-s − 7.51·26-s + 2.26·28-s + 8.10·29-s − 2.30·31-s + 6.50·32-s − 11.2·34-s − 3.96·35-s − 6.56·37-s + 1.82·38-s − 2.84·40-s − 7.75·41-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.668·4-s − 1.04·5-s + 0.639·7-s + 0.428·8-s + 1.35·10-s − 0.301·11-s + 1.14·13-s − 0.826·14-s − 1.22·16-s + 1.49·17-s − 0.229·19-s − 0.701·20-s + 0.389·22-s − 0.734·23-s + 0.100·25-s − 1.47·26-s + 0.427·28-s + 1.50·29-s − 0.413·31-s + 1.14·32-s − 1.93·34-s − 0.671·35-s − 1.07·37-s + 0.296·38-s − 0.449·40-s − 1.21·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1881\)    =    \(3^{2} \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(15.0198\)
Root analytic conductor: \(3.87554\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1881,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7225744816\)
\(L(\frac12)\) \(\approx\) \(0.7225744816\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 + 1.82T + 2T^{2} \)
5 \( 1 + 2.34T + 5T^{2} \)
7 \( 1 - 1.69T + 7T^{2} \)
13 \( 1 - 4.11T + 13T^{2} \)
17 \( 1 - 6.16T + 17T^{2} \)
23 \( 1 + 3.52T + 23T^{2} \)
29 \( 1 - 8.10T + 29T^{2} \)
31 \( 1 + 2.30T + 31T^{2} \)
37 \( 1 + 6.56T + 37T^{2} \)
41 \( 1 + 7.75T + 41T^{2} \)
43 \( 1 - 7.75T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 4.51T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 3.12T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 4.96T + 79T^{2} \)
83 \( 1 - 1.82T + 83T^{2} \)
89 \( 1 - 9.37T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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

−9.057069536077474394704367821141, −8.225341212758804531127500942415, −8.027716138457059395816028662087, −7.30671570520981387280144962846, −6.26922216990592444301311564758, −5.14310535193933174197090088145, −4.19166762113412063119720813498, −3.31294132823923443089918194155, −1.77986369409756526934564283482, −0.72824538954737085834307536004, 0.72824538954737085834307536004, 1.77986369409756526934564283482, 3.31294132823923443089918194155, 4.19166762113412063119720813498, 5.14310535193933174197090088145, 6.26922216990592444301311564758, 7.30671570520981387280144962846, 8.027716138457059395816028662087, 8.225341212758804531127500942415, 9.057069536077474394704367821141

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