Properties

Label 2-2151-1.1-c3-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.11·2-s + 1.72·4-s + 0.660·5-s − 24.6·7-s + 19.5·8-s − 2.05·10-s − 27.1·11-s + 16.3·13-s + 76.8·14-s − 74.8·16-s − 14.7·17-s − 127.·19-s + 1.13·20-s + 84.7·22-s − 159.·23-s − 124.·25-s − 51.0·26-s − 42.5·28-s − 229.·29-s + 244.·31-s + 76.8·32-s + 45.9·34-s − 16.2·35-s + 49.5·37-s + 399.·38-s + 12.9·40-s + 219.·41-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.215·4-s + 0.0590·5-s − 1.33·7-s + 0.864·8-s − 0.0651·10-s − 0.744·11-s + 0.349·13-s + 1.46·14-s − 1.16·16-s − 0.210·17-s − 1.54·19-s + 0.0127·20-s + 0.821·22-s − 1.44·23-s − 0.996·25-s − 0.384·26-s − 0.287·28-s − 1.46·29-s + 1.41·31-s + 0.424·32-s + 0.231·34-s − 0.0785·35-s + 0.220·37-s + 1.70·38-s + 0.0510·40-s + 0.835·41-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

\(L(2)\) \(\approx\) \(0.003305123053\)
\(L(\frac12)\) \(\approx\) \(0.003305123053\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.11T + 8T^{2} \)
5 \( 1 - 0.660T + 125T^{2} \)
7 \( 1 + 24.6T + 343T^{2} \)
11 \( 1 + 27.1T + 1.33e3T^{2} \)
13 \( 1 - 16.3T + 2.19e3T^{2} \)
17 \( 1 + 14.7T + 4.91e3T^{2} \)
19 \( 1 + 127.T + 6.85e3T^{2} \)
23 \( 1 + 159.T + 1.21e4T^{2} \)
29 \( 1 + 229.T + 2.43e4T^{2} \)
31 \( 1 - 244.T + 2.97e4T^{2} \)
37 \( 1 - 49.5T + 5.06e4T^{2} \)
41 \( 1 - 219.T + 6.89e4T^{2} \)
43 \( 1 - 10.3T + 7.95e4T^{2} \)
47 \( 1 + 337.T + 1.03e5T^{2} \)
53 \( 1 + 636.T + 1.48e5T^{2} \)
59 \( 1 + 154.T + 2.05e5T^{2} \)
61 \( 1 + 213.T + 2.26e5T^{2} \)
67 \( 1 + 116.T + 3.00e5T^{2} \)
71 \( 1 + 922.T + 3.57e5T^{2} \)
73 \( 1 - 529.T + 3.89e5T^{2} \)
79 \( 1 + 266.T + 4.93e5T^{2} \)
83 \( 1 + 1.51e3T + 5.71e5T^{2} \)
89 \( 1 - 96.6T + 7.04e5T^{2} \)
97 \( 1 + 1.22e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.724664079483917523103613341033, −8.095966908940755883851956230753, −7.43349152193028624438814174437, −6.37674404001005895256798974312, −5.93269695104014748045753749063, −4.56068787539105775306305843331, −3.81060660295066022593081843748, −2.60962992457781004391324410394, −1.65404595073461485553014211065, −0.03007384974866785182331538260, 0.03007384974866785182331538260, 1.65404595073461485553014211065, 2.60962992457781004391324410394, 3.81060660295066022593081843748, 4.56068787539105775306305843331, 5.93269695104014748045753749063, 6.37674404001005895256798974312, 7.43349152193028624438814174437, 8.095966908940755883851956230753, 8.724664079483917523103613341033

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