Properties

Label 2-39e2-1.1-c3-0-154
Degree $2$
Conductor $1521$
Sign $1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.04·2-s + 17.4·4-s + 20.1·5-s + 15.4·7-s + 47.7·8-s + 101.·10-s − 26.9·11-s + 77.8·14-s + 101.·16-s − 23.2·17-s − 45.0·19-s + 351.·20-s − 135.·22-s + 142.·23-s + 279.·25-s + 269.·28-s − 2.29·29-s − 37.7·31-s + 128.·32-s − 117.·34-s + 310.·35-s + 313.·37-s − 227.·38-s + 959.·40-s + 5.86·41-s − 360.·43-s − 470.·44-s + ⋯
L(s)  = 1  + 1.78·2-s + 2.18·4-s + 1.79·5-s + 0.833·7-s + 2.10·8-s + 3.20·10-s − 0.738·11-s + 1.48·14-s + 1.57·16-s − 0.331·17-s − 0.544·19-s + 3.92·20-s − 1.31·22-s + 1.28·23-s + 2.23·25-s + 1.81·28-s − 0.0146·29-s − 0.218·31-s + 0.708·32-s − 0.591·34-s + 1.49·35-s + 1.39·37-s − 0.970·38-s + 3.79·40-s + 0.0223·41-s − 1.27·43-s − 1.61·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(10.63359316\)
\(L(\frac12)\) \(\approx\) \(10.63359316\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 5.04T + 8T^{2} \)
5 \( 1 - 20.1T + 125T^{2} \)
7 \( 1 - 15.4T + 343T^{2} \)
11 \( 1 + 26.9T + 1.33e3T^{2} \)
17 \( 1 + 23.2T + 4.91e3T^{2} \)
19 \( 1 + 45.0T + 6.85e3T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 + 2.29T + 2.43e4T^{2} \)
31 \( 1 + 37.7T + 2.97e4T^{2} \)
37 \( 1 - 313.T + 5.06e4T^{2} \)
41 \( 1 - 5.86T + 6.89e4T^{2} \)
43 \( 1 + 360.T + 7.95e4T^{2} \)
47 \( 1 - 209.T + 1.03e5T^{2} \)
53 \( 1 + 276.T + 1.48e5T^{2} \)
59 \( 1 - 543.T + 2.05e5T^{2} \)
61 \( 1 - 205.T + 2.26e5T^{2} \)
67 \( 1 - 492.T + 3.00e5T^{2} \)
71 \( 1 + 826.T + 3.57e5T^{2} \)
73 \( 1 - 66.1T + 3.89e5T^{2} \)
79 \( 1 - 317.T + 4.93e5T^{2} \)
83 \( 1 + 141.T + 5.71e5T^{2} \)
89 \( 1 + 641.T + 7.04e5T^{2} \)
97 \( 1 + 1.11e3T + 9.12e5T^{2} \)
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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

−9.200735286289458063411558782524, −8.163489135623042579262171033336, −6.99204081696878097783024271956, −6.38383360868078794973024257561, −5.48395236519311653313846854671, −5.13602028965703558329245127034, −4.30396540466612687807627626285, −2.91862361283585892027380121044, −2.29495925968687375903221503636, −1.42430141688623934322855723612, 1.42430141688623934322855723612, 2.29495925968687375903221503636, 2.91862361283585892027380121044, 4.30396540466612687807627626285, 5.13602028965703558329245127034, 5.48395236519311653313846854671, 6.38383360868078794973024257561, 6.99204081696878097783024271956, 8.163489135623042579262171033336, 9.200735286289458063411558782524

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