Properties

Label 2-387-1.1-c7-0-80
Degree $2$
Conductor $387$
Sign $-1$
Analytic cond. $120.893$
Root an. cond. $10.9951$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.3·2-s + 172.·4-s + 122.·5-s + 247.·7-s − 777.·8-s − 2.13e3·10-s − 2.50e3·11-s + 1.37e3·13-s − 4.28e3·14-s − 8.63e3·16-s + 3.20e4·17-s + 1.02e4·19-s + 2.12e4·20-s + 4.34e4·22-s + 2.18e4·23-s − 6.30e4·25-s − 2.38e4·26-s + 4.26e4·28-s − 1.73e5·29-s + 1.60e4·31-s + 2.49e5·32-s − 5.56e5·34-s + 3.03e4·35-s − 7.56e4·37-s − 1.77e5·38-s − 9.55e4·40-s − 4.52e5·41-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.35·4-s + 0.439·5-s + 0.272·7-s − 0.536·8-s − 0.674·10-s − 0.567·11-s + 0.173·13-s − 0.417·14-s − 0.527·16-s + 1.58·17-s + 0.342·19-s + 0.593·20-s + 0.869·22-s + 0.374·23-s − 0.806·25-s − 0.266·26-s + 0.367·28-s − 1.32·29-s + 0.0967·31-s + 1.34·32-s − 2.42·34-s + 0.119·35-s − 0.245·37-s − 0.525·38-s − 0.236·40-s − 1.02·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(120.893\)
Root analytic conductor: \(10.9951\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 387,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + 7.95e4T \)
good2 \( 1 + 17.3T + 128T^{2} \)
5 \( 1 - 122.T + 7.81e4T^{2} \)
7 \( 1 - 247.T + 8.23e5T^{2} \)
11 \( 1 + 2.50e3T + 1.94e7T^{2} \)
13 \( 1 - 1.37e3T + 6.27e7T^{2} \)
17 \( 1 - 3.20e4T + 4.10e8T^{2} \)
19 \( 1 - 1.02e4T + 8.93e8T^{2} \)
23 \( 1 - 2.18e4T + 3.40e9T^{2} \)
29 \( 1 + 1.73e5T + 1.72e10T^{2} \)
31 \( 1 - 1.60e4T + 2.75e10T^{2} \)
37 \( 1 + 7.56e4T + 9.49e10T^{2} \)
41 \( 1 + 4.52e5T + 1.94e11T^{2} \)
47 \( 1 + 1.13e5T + 5.06e11T^{2} \)
53 \( 1 - 1.42e6T + 1.17e12T^{2} \)
59 \( 1 + 1.48e6T + 2.48e12T^{2} \)
61 \( 1 + 2.69e6T + 3.14e12T^{2} \)
67 \( 1 - 2.19e6T + 6.06e12T^{2} \)
71 \( 1 + 4.77e6T + 9.09e12T^{2} \)
73 \( 1 - 4.98e6T + 1.10e13T^{2} \)
79 \( 1 - 3.39e6T + 1.92e13T^{2} \)
83 \( 1 + 8.35e6T + 2.71e13T^{2} \)
89 \( 1 - 7.83e6T + 4.42e13T^{2} \)
97 \( 1 - 6.12e6T + 8.07e13T^{2} \)
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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

−9.729448984351888413309158348253, −8.840982866141610646390218045962, −7.88414963510288048676797777298, −7.35402168964710996586455126388, −6.04799884168863422201938327591, −5.04200135879814296431889773959, −3.37833076389183225276772455613, −2.03033029217755371205754674459, −1.17288161268328394555395624361, 0, 1.17288161268328394555395624361, 2.03033029217755371205754674459, 3.37833076389183225276772455613, 5.04200135879814296431889773959, 6.04799884168863422201938327591, 7.35402168964710996586455126388, 7.88414963510288048676797777298, 8.840982866141610646390218045962, 9.729448984351888413309158348253

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