Properties

Label 2-2151-1.1-c3-0-181
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  − 2.78·2-s − 0.253·4-s + 19.2·5-s + 27.5·7-s + 22.9·8-s − 53.5·10-s + 32.2·11-s + 10.3·13-s − 76.6·14-s − 61.9·16-s + 38.8·17-s + 137.·19-s − 4.88·20-s − 89.6·22-s − 158.·23-s + 245.·25-s − 28.7·26-s − 6.99·28-s − 30.8·29-s + 35.0·31-s − 11.4·32-s − 108.·34-s + 530.·35-s + 105.·37-s − 383.·38-s + 441.·40-s + 295.·41-s + ⋯
L(s)  = 1  − 0.984·2-s − 0.0317·4-s + 1.72·5-s + 1.48·7-s + 1.01·8-s − 1.69·10-s + 0.883·11-s + 0.220·13-s − 1.46·14-s − 0.967·16-s + 0.554·17-s + 1.66·19-s − 0.0545·20-s − 0.869·22-s − 1.43·23-s + 1.96·25-s − 0.217·26-s − 0.0472·28-s − 0.197·29-s + 0.202·31-s − 0.0634·32-s − 0.545·34-s + 2.55·35-s + 0.466·37-s − 1.63·38-s + 1.74·40-s + 1.12·41-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

\(L(2)\) \(\approx\) \(2.717067390\)
\(L(\frac12)\) \(\approx\) \(2.717067390\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.78T + 8T^{2} \)
5 \( 1 - 19.2T + 125T^{2} \)
7 \( 1 - 27.5T + 343T^{2} \)
11 \( 1 - 32.2T + 1.33e3T^{2} \)
13 \( 1 - 10.3T + 2.19e3T^{2} \)
17 \( 1 - 38.8T + 4.91e3T^{2} \)
19 \( 1 - 137.T + 6.85e3T^{2} \)
23 \( 1 + 158.T + 1.21e4T^{2} \)
29 \( 1 + 30.8T + 2.43e4T^{2} \)
31 \( 1 - 35.0T + 2.97e4T^{2} \)
37 \( 1 - 105.T + 5.06e4T^{2} \)
41 \( 1 - 295.T + 6.89e4T^{2} \)
43 \( 1 + 311.T + 7.95e4T^{2} \)
47 \( 1 + 122.T + 1.03e5T^{2} \)
53 \( 1 - 700.T + 1.48e5T^{2} \)
59 \( 1 + 548.T + 2.05e5T^{2} \)
61 \( 1 - 206.T + 2.26e5T^{2} \)
67 \( 1 - 352.T + 3.00e5T^{2} \)
71 \( 1 - 759.T + 3.57e5T^{2} \)
73 \( 1 + 587.T + 3.89e5T^{2} \)
79 \( 1 + 149.T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 1.04e3T + 7.04e5T^{2} \)
97 \( 1 + 1.80e3T + 9.12e5T^{2} \)
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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

−8.862564346557258184828207353775, −8.077028462148470304902250970870, −7.46277242814013632822479431395, −6.39213418407678085362818827772, −5.51334803093062347370453051481, −4.95459727861591273239705824342, −3.88025302749753763263594366174, −2.32108452608543222194322801369, −1.48491325846575055526299433536, −1.03797391223964977991131008989, 1.03797391223964977991131008989, 1.48491325846575055526299433536, 2.32108452608543222194322801369, 3.88025302749753763263594366174, 4.95459727861591273239705824342, 5.51334803093062347370453051481, 6.39213418407678085362818827772, 7.46277242814013632822479431395, 8.077028462148470304902250970870, 8.862564346557258184828207353775

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