Properties

Label 2-387-1.1-c7-0-113
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  + 20.3·2-s + 287.·4-s − 405.·5-s + 52.5·7-s + 3.24e3·8-s − 8.25e3·10-s − 2.54e3·11-s + 7.13e3·13-s + 1.07e3·14-s + 2.93e4·16-s + 2.25e4·17-s − 6.39e3·19-s − 1.16e5·20-s − 5.18e4·22-s − 9.98e4·23-s + 8.59e4·25-s + 1.45e5·26-s + 1.51e4·28-s − 5.60e3·29-s − 1.63e5·31-s + 1.83e5·32-s + 4.58e5·34-s − 2.13e4·35-s + 1.98e5·37-s − 1.30e5·38-s − 1.31e6·40-s − 7.42e5·41-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s − 1.44·5-s + 0.0579·7-s + 2.24·8-s − 2.61·10-s − 0.576·11-s + 0.901·13-s + 0.104·14-s + 1.79·16-s + 1.11·17-s − 0.213·19-s − 3.25·20-s − 1.03·22-s − 1.71·23-s + 1.10·25-s + 1.62·26-s + 0.130·28-s − 0.0427·29-s − 0.983·31-s + 0.987·32-s + 2.00·34-s − 0.0839·35-s + 0.643·37-s − 0.385·38-s − 3.24·40-s − 1.68·41-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 \( 1 - 20.3T + 128T^{2} \)
5 \( 1 + 405.T + 7.81e4T^{2} \)
7 \( 1 - 52.5T + 8.23e5T^{2} \)
11 \( 1 + 2.54e3T + 1.94e7T^{2} \)
13 \( 1 - 7.13e3T + 6.27e7T^{2} \)
17 \( 1 - 2.25e4T + 4.10e8T^{2} \)
19 \( 1 + 6.39e3T + 8.93e8T^{2} \)
23 \( 1 + 9.98e4T + 3.40e9T^{2} \)
29 \( 1 + 5.60e3T + 1.72e10T^{2} \)
31 \( 1 + 1.63e5T + 2.75e10T^{2} \)
37 \( 1 - 1.98e5T + 9.49e10T^{2} \)
41 \( 1 + 7.42e5T + 1.94e11T^{2} \)
47 \( 1 + 7.86e5T + 5.06e11T^{2} \)
53 \( 1 + 2.08e6T + 1.17e12T^{2} \)
59 \( 1 + 6.07e5T + 2.48e12T^{2} \)
61 \( 1 - 2.59e6T + 3.14e12T^{2} \)
67 \( 1 + 7.55e5T + 6.06e12T^{2} \)
71 \( 1 - 3.83e5T + 9.09e12T^{2} \)
73 \( 1 - 2.94e6T + 1.10e13T^{2} \)
79 \( 1 + 7.28e6T + 1.92e13T^{2} \)
83 \( 1 - 1.91e6T + 2.71e13T^{2} \)
89 \( 1 + 1.05e7T + 4.42e13T^{2} \)
97 \( 1 - 1.18e7T + 8.07e13T^{2} \)
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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

−10.07981361862936816065929508799, −8.246230096744988742415763370547, −7.69183966980252370143537388559, −6.58006429626823545390048330873, −5.62586420181178725252706301619, −4.64063947216201107373650154599, −3.72521528130531241163976466955, −3.21184038470250593399170772530, −1.71896771378764195089523969720, 0, 1.71896771378764195089523969720, 3.21184038470250593399170772530, 3.72521528130531241163976466955, 4.64063947216201107373650154599, 5.62586420181178725252706301619, 6.58006429626823545390048330873, 7.69183966980252370143537388559, 8.246230096744988742415763370547, 10.07981361862936816065929508799

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