Properties

Label 2-2151-1.1-c1-0-33
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
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  − 0.685·2-s − 1.52·4-s + 0.233·5-s + 1.65·7-s + 2.42·8-s − 0.159·10-s + 5.18·11-s + 4.45·13-s − 1.13·14-s + 1.39·16-s + 1.66·17-s − 0.667·19-s − 0.356·20-s − 3.55·22-s + 6.83·23-s − 4.94·25-s − 3.05·26-s − 2.52·28-s − 0.562·29-s − 10.4·31-s − 5.80·32-s − 1.13·34-s + 0.385·35-s + 5.83·37-s + 0.457·38-s + 0.564·40-s − 8.92·41-s + ⋯
L(s)  = 1  − 0.484·2-s − 0.764·4-s + 0.104·5-s + 0.624·7-s + 0.855·8-s − 0.0505·10-s + 1.56·11-s + 1.23·13-s − 0.302·14-s + 0.349·16-s + 0.403·17-s − 0.153·19-s − 0.0797·20-s − 0.757·22-s + 1.42·23-s − 0.989·25-s − 0.599·26-s − 0.477·28-s − 0.104·29-s − 1.86·31-s − 1.02·32-s − 0.195·34-s + 0.0651·35-s + 0.959·37-s + 0.0742·38-s + 0.0892·40-s − 1.39·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513292988\)
\(L(\frac12)\) \(\approx\) \(1.513292988\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 + 0.685T + 2T^{2} \)
5 \( 1 - 0.233T + 5T^{2} \)
7 \( 1 - 1.65T + 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
13 \( 1 - 4.45T + 13T^{2} \)
17 \( 1 - 1.66T + 17T^{2} \)
19 \( 1 + 0.667T + 19T^{2} \)
23 \( 1 - 6.83T + 23T^{2} \)
29 \( 1 + 0.562T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 5.83T + 37T^{2} \)
41 \( 1 + 8.92T + 41T^{2} \)
43 \( 1 - 6.75T + 43T^{2} \)
47 \( 1 - 1.55T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 3.29T + 59T^{2} \)
61 \( 1 - 1.95T + 61T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 - 0.604T + 71T^{2} \)
73 \( 1 + 2.07T + 73T^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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

−9.083595647568863010455556714813, −8.500076718252752874661811114174, −7.70042242493246863173806068099, −6.85045298071149364049419028414, −5.88998127900124176228392863546, −5.08711902654035633505592476192, −4.05612633310847878687229026268, −3.55895622129760722569573964275, −1.75643748464247847774597296333, −0.975506007691015832262219976197, 0.975506007691015832262219976197, 1.75643748464247847774597296333, 3.55895622129760722569573964275, 4.05612633310847878687229026268, 5.08711902654035633505592476192, 5.88998127900124176228392863546, 6.85045298071149364049419028414, 7.70042242493246863173806068099, 8.500076718252752874661811114174, 9.083595647568863010455556714813

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