Properties

Label 2-39e2-1.1-c3-0-151
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.15·2-s − 6.67·4-s + 21.2·5-s − 31.2·7-s − 16.8·8-s + 24.3·10-s + 17.3·11-s − 35.9·14-s + 33.9·16-s + 89.1·17-s − 80.6·19-s − 141.·20-s + 19.9·22-s − 149.·23-s + 324.·25-s + 208.·28-s + 6.30·29-s + 78.3·31-s + 174.·32-s + 102.·34-s − 662.·35-s + 39.2·37-s − 92.7·38-s − 358.·40-s + 330.·41-s − 198.·43-s − 116.·44-s + ⋯
L(s)  = 1  + 0.406·2-s − 0.834·4-s + 1.89·5-s − 1.68·7-s − 0.746·8-s + 0.771·10-s + 0.476·11-s − 0.686·14-s + 0.530·16-s + 1.27·17-s − 0.973·19-s − 1.58·20-s + 0.193·22-s − 1.35·23-s + 2.59·25-s + 1.40·28-s + 0.0403·29-s + 0.453·31-s + 0.962·32-s + 0.517·34-s − 3.20·35-s + 0.174·37-s − 0.396·38-s − 1.41·40-s + 1.25·41-s − 0.705·43-s − 0.397·44-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: \(1\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.15T + 8T^{2} \)
5 \( 1 - 21.2T + 125T^{2} \)
7 \( 1 + 31.2T + 343T^{2} \)
11 \( 1 - 17.3T + 1.33e3T^{2} \)
17 \( 1 - 89.1T + 4.91e3T^{2} \)
19 \( 1 + 80.6T + 6.85e3T^{2} \)
23 \( 1 + 149.T + 1.21e4T^{2} \)
29 \( 1 - 6.30T + 2.43e4T^{2} \)
31 \( 1 - 78.3T + 2.97e4T^{2} \)
37 \( 1 - 39.2T + 5.06e4T^{2} \)
41 \( 1 - 330.T + 6.89e4T^{2} \)
43 \( 1 + 198.T + 7.95e4T^{2} \)
47 \( 1 + 246.T + 1.03e5T^{2} \)
53 \( 1 + 600.T + 1.48e5T^{2} \)
59 \( 1 + 709.T + 2.05e5T^{2} \)
61 \( 1 + 472.T + 2.26e5T^{2} \)
67 \( 1 + 331.T + 3.00e5T^{2} \)
71 \( 1 + 472.T + 3.57e5T^{2} \)
73 \( 1 - 651.T + 3.89e5T^{2} \)
79 \( 1 - 240.T + 4.93e5T^{2} \)
83 \( 1 - 538.T + 5.71e5T^{2} \)
89 \( 1 + 673.T + 7.04e5T^{2} \)
97 \( 1 + 468.T + 9.12e5T^{2} \)
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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

−9.107426933386287721249036899801, −8.032340734886527622859822512167, −6.54874928418066696363389705856, −6.15102664568799261221551391076, −5.63311628890535876563179953734, −4.53427556836427995718113716866, −3.45462581822654530251868319032, −2.67671260919520335526172360282, −1.38136089240566019292117426859, 0, 1.38136089240566019292117426859, 2.67671260919520335526172360282, 3.45462581822654530251868319032, 4.53427556836427995718113716866, 5.63311628890535876563179953734, 6.15102664568799261221551391076, 6.54874928418066696363389705856, 8.032340734886527622859822512167, 9.107426933386287721249036899801

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