Properties

Label 2-387-1.1-c7-0-109
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  + 15.1·2-s + 101.·4-s − 210.·5-s + 1.10e3·7-s − 399.·8-s − 3.19e3·10-s + 3.13e3·11-s − 5.74e3·13-s + 1.66e4·14-s − 1.90e4·16-s − 2.72e4·17-s + 5.46e4·19-s − 2.14e4·20-s + 4.75e4·22-s − 1.20e4·23-s − 3.36e4·25-s − 8.70e4·26-s + 1.11e5·28-s − 1.32e5·29-s + 2.81e5·31-s − 2.37e5·32-s − 4.12e5·34-s − 2.32e5·35-s − 2.14e5·37-s + 8.28e5·38-s + 8.42e4·40-s − 3.02e5·41-s + ⋯
L(s)  = 1  + 1.33·2-s + 0.794·4-s − 0.754·5-s + 1.21·7-s − 0.275·8-s − 1.01·10-s + 0.710·11-s − 0.725·13-s + 1.62·14-s − 1.16·16-s − 1.34·17-s + 1.82·19-s − 0.599·20-s + 0.951·22-s − 0.206·23-s − 0.430·25-s − 0.971·26-s + 0.963·28-s − 1.00·29-s + 1.69·31-s − 1.28·32-s − 1.80·34-s − 0.915·35-s − 0.696·37-s + 2.44·38-s + 0.208·40-s − 0.684·41-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 \( 1 - 15.1T + 128T^{2} \)
5 \( 1 + 210.T + 7.81e4T^{2} \)
7 \( 1 - 1.10e3T + 8.23e5T^{2} \)
11 \( 1 - 3.13e3T + 1.94e7T^{2} \)
13 \( 1 + 5.74e3T + 6.27e7T^{2} \)
17 \( 1 + 2.72e4T + 4.10e8T^{2} \)
19 \( 1 - 5.46e4T + 8.93e8T^{2} \)
23 \( 1 + 1.20e4T + 3.40e9T^{2} \)
29 \( 1 + 1.32e5T + 1.72e10T^{2} \)
31 \( 1 - 2.81e5T + 2.75e10T^{2} \)
37 \( 1 + 2.14e5T + 9.49e10T^{2} \)
41 \( 1 + 3.02e5T + 1.94e11T^{2} \)
47 \( 1 + 2.35e5T + 5.06e11T^{2} \)
53 \( 1 + 5.50e4T + 1.17e12T^{2} \)
59 \( 1 + 1.50e5T + 2.48e12T^{2} \)
61 \( 1 + 1.65e6T + 3.14e12T^{2} \)
67 \( 1 - 1.76e6T + 6.06e12T^{2} \)
71 \( 1 + 3.57e6T + 9.09e12T^{2} \)
73 \( 1 + 4.30e5T + 1.10e13T^{2} \)
79 \( 1 + 7.44e6T + 1.92e13T^{2} \)
83 \( 1 + 5.50e6T + 2.71e13T^{2} \)
89 \( 1 - 3.53e6T + 4.42e13T^{2} \)
97 \( 1 + 2.08e6T + 8.07e13T^{2} \)
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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

−9.690600115675773465662431306357, −8.636951054149408600323470508157, −7.63388853987088442873851877117, −6.73477514700548131028992877987, −5.48124447414029518623817560068, −4.67278610920579135852835291770, −4.01555794915144913460382324067, −2.88353758230972069285664864676, −1.58483051618166992724270485627, 0, 1.58483051618166992724270485627, 2.88353758230972069285664864676, 4.01555794915144913460382324067, 4.67278610920579135852835291770, 5.48124447414029518623817560068, 6.73477514700548131028992877987, 7.63388853987088442873851877117, 8.636951054149408600323470508157, 9.690600115675773465662431306357

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