Properties

Label 2-387-1.1-c7-0-59
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  − 22.1·2-s + 364.·4-s − 122.·5-s − 90.8·7-s − 5.23e3·8-s + 2.71e3·10-s − 3.41e3·11-s − 1.12e4·13-s + 2.01e3·14-s + 6.95e4·16-s − 9.72e3·17-s + 2.53e4·19-s − 4.46e4·20-s + 7.56e4·22-s + 5.27e4·23-s − 6.31e4·25-s + 2.50e5·26-s − 3.30e4·28-s + 2.16e5·29-s + 1.42e5·31-s − 8.72e5·32-s + 2.15e5·34-s + 1.11e4·35-s − 7.42e4·37-s − 5.62e5·38-s + 6.41e5·40-s + 2.73e5·41-s + ⋯
L(s)  = 1  − 1.96·2-s + 2.84·4-s − 0.438·5-s − 0.100·7-s − 3.61·8-s + 0.859·10-s − 0.772·11-s − 1.42·13-s + 0.196·14-s + 4.24·16-s − 0.479·17-s + 0.847·19-s − 1.24·20-s + 1.51·22-s + 0.904·23-s − 0.807·25-s + 2.79·26-s − 0.284·28-s + 1.65·29-s + 0.856·31-s − 4.70·32-s + 0.940·34-s + 0.0438·35-s − 0.241·37-s − 1.66·38-s + 1.58·40-s + 0.618·41-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 \( 1 + 22.1T + 128T^{2} \)
5 \( 1 + 122.T + 7.81e4T^{2} \)
7 \( 1 + 90.8T + 8.23e5T^{2} \)
11 \( 1 + 3.41e3T + 1.94e7T^{2} \)
13 \( 1 + 1.12e4T + 6.27e7T^{2} \)
17 \( 1 + 9.72e3T + 4.10e8T^{2} \)
19 \( 1 - 2.53e4T + 8.93e8T^{2} \)
23 \( 1 - 5.27e4T + 3.40e9T^{2} \)
29 \( 1 - 2.16e5T + 1.72e10T^{2} \)
31 \( 1 - 1.42e5T + 2.75e10T^{2} \)
37 \( 1 + 7.42e4T + 9.49e10T^{2} \)
41 \( 1 - 2.73e5T + 1.94e11T^{2} \)
47 \( 1 - 1.12e5T + 5.06e11T^{2} \)
53 \( 1 + 1.41e6T + 1.17e12T^{2} \)
59 \( 1 - 1.44e6T + 2.48e12T^{2} \)
61 \( 1 - 3.19e6T + 3.14e12T^{2} \)
67 \( 1 + 3.52e6T + 6.06e12T^{2} \)
71 \( 1 - 3.27e5T + 9.09e12T^{2} \)
73 \( 1 - 2.80e6T + 1.10e13T^{2} \)
79 \( 1 - 3.35e6T + 1.92e13T^{2} \)
83 \( 1 + 7.10e6T + 2.71e13T^{2} \)
89 \( 1 - 1.32e6T + 4.42e13T^{2} \)
97 \( 1 - 1.29e7T + 8.07e13T^{2} \)
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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

−9.775560395048408296003601731869, −8.760015253242280624819126653720, −7.899034932194663277461274274579, −7.30698404050193698253159408263, −6.40179178362389475997125290876, −5.00747727091059321923346366678, −3.05461762310461832453955410451, −2.27487809424816993348349763588, −0.894222761137565858921941206317, 0, 0.894222761137565858921941206317, 2.27487809424816993348349763588, 3.05461762310461832453955410451, 5.00747727091059321923346366678, 6.40179178362389475997125290876, 7.30698404050193698253159408263, 7.899034932194663277461274274579, 8.760015253242280624819126653720, 9.775560395048408296003601731869

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