Properties

Label 2-2151-1.1-c3-0-14
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  + 2.03·2-s − 3.87·4-s − 11.5·5-s − 13.0·7-s − 24.1·8-s − 23.5·10-s + 22.2·11-s − 19.5·13-s − 26.5·14-s − 18.0·16-s − 90.8·17-s − 85.3·19-s + 44.8·20-s + 45.1·22-s + 3.09·23-s + 8.92·25-s − 39.8·26-s + 50.6·28-s − 58.6·29-s − 237.·31-s + 156.·32-s − 184.·34-s + 151.·35-s − 113.·37-s − 173.·38-s + 279.·40-s − 375.·41-s + ⋯
L(s)  = 1  + 0.718·2-s − 0.484·4-s − 1.03·5-s − 0.706·7-s − 1.06·8-s − 0.743·10-s + 0.609·11-s − 0.418·13-s − 0.507·14-s − 0.281·16-s − 1.29·17-s − 1.03·19-s + 0.501·20-s + 0.437·22-s + 0.0280·23-s + 0.0714·25-s − 0.300·26-s + 0.342·28-s − 0.375·29-s − 1.37·31-s + 0.863·32-s − 0.931·34-s + 0.731·35-s − 0.506·37-s − 0.739·38-s + 1.10·40-s − 1.43·41-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

\(L(2)\) \(\approx\) \(0.1453002025\)
\(L(\frac12)\) \(\approx\) \(0.1453002025\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.03T + 8T^{2} \)
5 \( 1 + 11.5T + 125T^{2} \)
7 \( 1 + 13.0T + 343T^{2} \)
11 \( 1 - 22.2T + 1.33e3T^{2} \)
13 \( 1 + 19.5T + 2.19e3T^{2} \)
17 \( 1 + 90.8T + 4.91e3T^{2} \)
19 \( 1 + 85.3T + 6.85e3T^{2} \)
23 \( 1 - 3.09T + 1.21e4T^{2} \)
29 \( 1 + 58.6T + 2.43e4T^{2} \)
31 \( 1 + 237.T + 2.97e4T^{2} \)
37 \( 1 + 113.T + 5.06e4T^{2} \)
41 \( 1 + 375.T + 6.89e4T^{2} \)
43 \( 1 + 348.T + 7.95e4T^{2} \)
47 \( 1 + 501.T + 1.03e5T^{2} \)
53 \( 1 + 174.T + 1.48e5T^{2} \)
59 \( 1 - 758.T + 2.05e5T^{2} \)
61 \( 1 + 13.7T + 2.26e5T^{2} \)
67 \( 1 - 92.1T + 3.00e5T^{2} \)
71 \( 1 - 746.T + 3.57e5T^{2} \)
73 \( 1 - 131.T + 3.89e5T^{2} \)
79 \( 1 + 49.9T + 4.93e5T^{2} \)
83 \( 1 - 21.6T + 5.71e5T^{2} \)
89 \( 1 + 188.T + 7.04e5T^{2} \)
97 \( 1 + 667.T + 9.12e5T^{2} \)
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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

−8.691809838142761056375191489538, −8.084960511713812841843378483692, −6.83802257787085613309848252457, −6.53078605146361478809767603841, −5.34145262836917362768781393308, −4.57119674114890585483845561512, −3.80668547904707734887286283231, −3.32851293635240617382456462988, −1.99377398825330115873908763997, −0.14650584807786127433265587212, 0.14650584807786127433265587212, 1.99377398825330115873908763997, 3.32851293635240617382456462988, 3.80668547904707734887286283231, 4.57119674114890585483845561512, 5.34145262836917362768781393308, 6.53078605146361478809767603841, 6.83802257787085613309848252457, 8.084960511713812841843378483692, 8.691809838142761056375191489538

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