Properties

Label 2-387-1.1-c7-0-43
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  − 0.315·2-s − 127.·4-s − 531.·5-s − 349.·7-s + 80.7·8-s + 167.·10-s − 3.33e3·11-s − 8.58e3·13-s + 110.·14-s + 1.63e4·16-s + 2.57e4·17-s + 1.69e4·19-s + 6.79e4·20-s + 1.05e3·22-s − 8.07e4·23-s + 2.04e5·25-s + 2.70e3·26-s + 4.47e4·28-s + 6.77e4·29-s + 1.88e4·31-s − 1.54e4·32-s − 8.13e3·34-s + 1.85e5·35-s − 1.79e5·37-s − 5.36e3·38-s − 4.29e4·40-s + 6.11e5·41-s + ⋯
L(s)  = 1  − 0.0279·2-s − 0.999·4-s − 1.90·5-s − 0.385·7-s + 0.0557·8-s + 0.0530·10-s − 0.754·11-s − 1.08·13-s + 0.0107·14-s + 0.997·16-s + 1.27·17-s + 0.568·19-s + 1.90·20-s + 0.0210·22-s − 1.38·23-s + 2.61·25-s + 0.0302·26-s + 0.385·28-s + 0.515·29-s + 0.113·31-s − 0.0836·32-s − 0.0355·34-s + 0.733·35-s − 0.582·37-s − 0.0158·38-s − 0.106·40-s + 1.38·41-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 \( 1 + 0.315T + 128T^{2} \)
5 \( 1 + 531.T + 7.81e4T^{2} \)
7 \( 1 + 349.T + 8.23e5T^{2} \)
11 \( 1 + 3.33e3T + 1.94e7T^{2} \)
13 \( 1 + 8.58e3T + 6.27e7T^{2} \)
17 \( 1 - 2.57e4T + 4.10e8T^{2} \)
19 \( 1 - 1.69e4T + 8.93e8T^{2} \)
23 \( 1 + 8.07e4T + 3.40e9T^{2} \)
29 \( 1 - 6.77e4T + 1.72e10T^{2} \)
31 \( 1 - 1.88e4T + 2.75e10T^{2} \)
37 \( 1 + 1.79e5T + 9.49e10T^{2} \)
41 \( 1 - 6.11e5T + 1.94e11T^{2} \)
47 \( 1 - 1.18e6T + 5.06e11T^{2} \)
53 \( 1 + 5.38e5T + 1.17e12T^{2} \)
59 \( 1 - 1.96e6T + 2.48e12T^{2} \)
61 \( 1 - 2.10e6T + 3.14e12T^{2} \)
67 \( 1 + 8.13e5T + 6.06e12T^{2} \)
71 \( 1 + 1.76e6T + 9.09e12T^{2} \)
73 \( 1 + 3.47e6T + 1.10e13T^{2} \)
79 \( 1 + 3.68e6T + 1.92e13T^{2} \)
83 \( 1 + 3.77e6T + 2.71e13T^{2} \)
89 \( 1 - 2.83e6T + 4.42e13T^{2} \)
97 \( 1 - 4.40e6T + 8.07e13T^{2} \)
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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

−9.754267370582622068228642307124, −8.563159666444096914187153745879, −7.81029417348879053138146110606, −7.31730089353232018722754659630, −5.59894397191645951682952637603, −4.62486375025609859358135205077, −3.81015326517505351336866206635, −2.90927867482978221681082785706, −0.76805713188906949929603809059, 0, 0.76805713188906949929603809059, 2.90927867482978221681082785706, 3.81015326517505351336866206635, 4.62486375025609859358135205077, 5.59894397191645951682952637603, 7.31730089353232018722754659630, 7.81029417348879053138146110606, 8.563159666444096914187153745879, 9.754267370582622068228642307124

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