Properties

Label 2-39e2-1.1-c3-0-119
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  + 3.85·2-s + 6.86·4-s + 19.5·5-s + 6.26·7-s − 4.36·8-s + 75.3·10-s + 27.2·11-s + 24.1·14-s − 71.7·16-s + 30.8·17-s − 127.·19-s + 134.·20-s + 105.·22-s + 84.3·23-s + 256.·25-s + 43.0·28-s + 272.·29-s + 166.·31-s − 241.·32-s + 118.·34-s + 122.·35-s + 198.·37-s − 492.·38-s − 85.2·40-s − 160.·41-s + 158.·43-s + 187.·44-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.858·4-s + 1.74·5-s + 0.338·7-s − 0.192·8-s + 2.38·10-s + 0.746·11-s + 0.461·14-s − 1.12·16-s + 0.440·17-s − 1.54·19-s + 1.49·20-s + 1.01·22-s + 0.764·23-s + 2.05·25-s + 0.290·28-s + 1.74·29-s + 0.963·31-s − 1.33·32-s + 0.599·34-s + 0.590·35-s + 0.880·37-s − 2.10·38-s − 0.337·40-s − 0.610·41-s + 0.563·43-s + 0.641·44-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

\(L(2)\) \(\approx\) \(7.090585938\)
\(L(\frac12)\) \(\approx\) \(7.090585938\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.85T + 8T^{2} \)
5 \( 1 - 19.5T + 125T^{2} \)
7 \( 1 - 6.26T + 343T^{2} \)
11 \( 1 - 27.2T + 1.33e3T^{2} \)
17 \( 1 - 30.8T + 4.91e3T^{2} \)
19 \( 1 + 127.T + 6.85e3T^{2} \)
23 \( 1 - 84.3T + 1.21e4T^{2} \)
29 \( 1 - 272.T + 2.43e4T^{2} \)
31 \( 1 - 166.T + 2.97e4T^{2} \)
37 \( 1 - 198.T + 5.06e4T^{2} \)
41 \( 1 + 160.T + 6.89e4T^{2} \)
43 \( 1 - 158.T + 7.95e4T^{2} \)
47 \( 1 - 305.T + 1.03e5T^{2} \)
53 \( 1 - 356.T + 1.48e5T^{2} \)
59 \( 1 - 470.T + 2.05e5T^{2} \)
61 \( 1 + 171.T + 2.26e5T^{2} \)
67 \( 1 + 1.02e3T + 3.00e5T^{2} \)
71 \( 1 - 188.T + 3.57e5T^{2} \)
73 \( 1 + 959.T + 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 - 105.T + 5.71e5T^{2} \)
89 \( 1 - 649.T + 7.04e5T^{2} \)
97 \( 1 + 707.T + 9.12e5T^{2} \)
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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

−9.056408634901036443012566316479, −8.537192656672218266624933728337, −6.98930136071189772415526789637, −6.25851941313711178118809708893, −5.87417981942473984814014811701, −4.86578790436755619937895737924, −4.31575649839361882174505172912, −2.98277169843247097047485881714, −2.26138110203357270172048365855, −1.14788577304133737792447267803, 1.14788577304133737792447267803, 2.26138110203357270172048365855, 2.98277169843247097047485881714, 4.31575649839361882174505172912, 4.86578790436755619937895737924, 5.87417981942473984814014811701, 6.25851941313711178118809708893, 6.98930136071189772415526789637, 8.537192656672218266624933728337, 9.056408634901036443012566316479

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