Properties

Label 2-387-1.1-c9-0-37
Degree $2$
Conductor $387$
Sign $1$
Analytic cond. $199.318$
Root an. cond. $14.1180$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 43.0·2-s + 1.34e3·4-s + 1.33e3·5-s − 743.·7-s − 3.58e4·8-s − 5.73e4·10-s + 6.08e4·11-s − 1.04e5·13-s + 3.20e4·14-s + 8.58e5·16-s + 3.44e5·17-s − 6.26e5·19-s + 1.79e6·20-s − 2.62e6·22-s − 1.13e6·23-s − 1.81e5·25-s + 4.48e6·26-s − 1.00e6·28-s − 3.51e6·29-s − 6.97e6·31-s − 1.86e7·32-s − 1.48e7·34-s − 9.89e5·35-s + 7.60e6·37-s + 2.70e7·38-s − 4.77e7·40-s − 3.04e7·41-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.62·4-s + 0.952·5-s − 0.117·7-s − 3.09·8-s − 1.81·10-s + 1.25·11-s − 1.01·13-s + 0.222·14-s + 3.27·16-s + 1.00·17-s − 1.10·19-s + 2.50·20-s − 2.38·22-s − 0.842·23-s − 0.0929·25-s + 1.92·26-s − 0.307·28-s − 0.922·29-s − 1.35·31-s − 3.13·32-s − 1.90·34-s − 0.111·35-s + 0.666·37-s + 2.10·38-s − 2.95·40-s − 1.68·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(199.318\)
Root analytic conductor: \(14.1180\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.8545666057\)
\(L(\frac12)\) \(\approx\) \(0.8545666057\)
\(L(\frac{11}{2})\) 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 \( 1 + 3.41e6T \)
good2 \( 1 + 43.0T + 512T^{2} \)
5 \( 1 - 1.33e3T + 1.95e6T^{2} \)
7 \( 1 + 743.T + 4.03e7T^{2} \)
11 \( 1 - 6.08e4T + 2.35e9T^{2} \)
13 \( 1 + 1.04e5T + 1.06e10T^{2} \)
17 \( 1 - 3.44e5T + 1.18e11T^{2} \)
19 \( 1 + 6.26e5T + 3.22e11T^{2} \)
23 \( 1 + 1.13e6T + 1.80e12T^{2} \)
29 \( 1 + 3.51e6T + 1.45e13T^{2} \)
31 \( 1 + 6.97e6T + 2.64e13T^{2} \)
37 \( 1 - 7.60e6T + 1.29e14T^{2} \)
41 \( 1 + 3.04e7T + 3.27e14T^{2} \)
47 \( 1 + 3.57e7T + 1.11e15T^{2} \)
53 \( 1 - 9.41e7T + 3.29e15T^{2} \)
59 \( 1 - 1.25e8T + 8.66e15T^{2} \)
61 \( 1 - 1.62e8T + 1.16e16T^{2} \)
67 \( 1 - 1.76e8T + 2.72e16T^{2} \)
71 \( 1 - 7.75e7T + 4.58e16T^{2} \)
73 \( 1 + 4.14e8T + 5.88e16T^{2} \)
79 \( 1 - 3.87e8T + 1.19e17T^{2} \)
83 \( 1 + 1.61e7T + 1.86e17T^{2} \)
89 \( 1 - 1.53e8T + 3.50e17T^{2} \)
97 \( 1 - 1.02e9T + 7.60e17T^{2} \)
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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

−9.857754976806218632822506769947, −9.058416139486146050299110186216, −8.197041089593507225688246751098, −7.17090005517898504931079618238, −6.44303223482752625824587838237, −5.53533241615143394963442539313, −3.60311775112092644609198183048, −2.15014046606681922743458737849, −1.71590774198702733165711745833, −0.50942267096239345314348907280, 0.50942267096239345314348907280, 1.71590774198702733165711745833, 2.15014046606681922743458737849, 3.60311775112092644609198183048, 5.53533241615143394963442539313, 6.44303223482752625824587838237, 7.17090005517898504931079618238, 8.197041089593507225688246751098, 9.058416139486146050299110186216, 9.857754976806218632822506769947

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