Properties

Label 2-2151-1.1-c3-0-39
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $126.913$
Root an. cond. $11.2655$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.96·2-s + 0.788·4-s + 20.3·5-s − 25.7·7-s + 21.3·8-s − 60.1·10-s − 49.6·11-s + 51.2·13-s + 76.1·14-s − 69.6·16-s − 127.·17-s − 97.6·19-s + 16.0·20-s + 147.·22-s − 175.·23-s + 287.·25-s − 151.·26-s − 20.2·28-s − 16.4·29-s − 289.·31-s + 35.5·32-s + 378.·34-s − 521.·35-s + 86.9·37-s + 289.·38-s + 434.·40-s − 88.5·41-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.0985·4-s + 1.81·5-s − 1.38·7-s + 0.944·8-s − 1.90·10-s − 1.35·11-s + 1.09·13-s + 1.45·14-s − 1.08·16-s − 1.82·17-s − 1.17·19-s + 0.178·20-s + 1.42·22-s − 1.59·23-s + 2.29·25-s − 1.14·26-s − 0.136·28-s − 0.105·29-s − 1.67·31-s + 0.196·32-s + 1.90·34-s − 2.52·35-s + 0.386·37-s + 1.23·38-s + 1.71·40-s − 0.337·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(126.913\)
Root analytic conductor: \(11.2655\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5652218001\)
\(L(\frac12)\) \(\approx\) \(0.5652218001\)
\(L(\frac{5}{2})\) 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 \( 1 + 239T \)
good2 \( 1 + 2.96T + 8T^{2} \)
5 \( 1 - 20.3T + 125T^{2} \)
7 \( 1 + 25.7T + 343T^{2} \)
11 \( 1 + 49.6T + 1.33e3T^{2} \)
13 \( 1 - 51.2T + 2.19e3T^{2} \)
17 \( 1 + 127.T + 4.91e3T^{2} \)
19 \( 1 + 97.6T + 6.85e3T^{2} \)
23 \( 1 + 175.T + 1.21e4T^{2} \)
29 \( 1 + 16.4T + 2.43e4T^{2} \)
31 \( 1 + 289.T + 2.97e4T^{2} \)
37 \( 1 - 86.9T + 5.06e4T^{2} \)
41 \( 1 + 88.5T + 6.89e4T^{2} \)
43 \( 1 + 346.T + 7.95e4T^{2} \)
47 \( 1 - 346.T + 1.03e5T^{2} \)
53 \( 1 + 234.T + 1.48e5T^{2} \)
59 \( 1 - 427.T + 2.05e5T^{2} \)
61 \( 1 - 170.T + 2.26e5T^{2} \)
67 \( 1 - 724.T + 3.00e5T^{2} \)
71 \( 1 + 29.5T + 3.57e5T^{2} \)
73 \( 1 + 1.20e3T + 3.89e5T^{2} \)
79 \( 1 - 1.20e3T + 4.93e5T^{2} \)
83 \( 1 - 191.T + 5.71e5T^{2} \)
89 \( 1 - 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 58.7T + 9.12e5T^{2} \)
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

−8.877705086076100655638849894706, −8.335155383150535706052600555592, −7.12162354487654802856055642652, −6.33795019665408062234708113308, −5.89787943329636133250509751682, −4.85457004542000663516640623405, −3.69839625891451147330547092690, −2.29449261611501478818301680869, −1.93619076213978370805313021849, −0.37862981426351275850846076241, 0.37862981426351275850846076241, 1.93619076213978370805313021849, 2.29449261611501478818301680869, 3.69839625891451147330547092690, 4.85457004542000663516640623405, 5.89787943329636133250509751682, 6.33795019665408062234708113308, 7.12162354487654802856055642652, 8.335155383150535706052600555592, 8.877705086076100655638849894706

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