Properties

Label 2-39e2-1.1-c3-0-95
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  − 2.64·2-s − 0.999·4-s + 10.5·5-s + 22·7-s + 23.8·8-s − 28.0·10-s + 5.29·11-s − 58.2·14-s − 55.0·16-s + 116.·17-s + 126·19-s − 10.5·20-s − 14.0·22-s + 31.7·23-s − 12.9·25-s − 21.9·28-s + 52.9·29-s + 182·31-s − 44.9·32-s − 308·34-s + 232.·35-s + 86·37-s − 333.·38-s + 252.·40-s + 444.·41-s + 96·43-s − 5.29·44-s + ⋯
L(s)  = 1  − 0.935·2-s − 0.124·4-s + 0.946·5-s + 1.18·7-s + 1.05·8-s − 0.885·10-s + 0.145·11-s − 1.11·14-s − 0.859·16-s + 1.66·17-s + 1.52·19-s − 0.118·20-s − 0.135·22-s + 0.287·23-s − 0.103·25-s − 0.148·28-s + 0.338·29-s + 1.05·31-s − 0.248·32-s − 1.55·34-s + 1.12·35-s + 0.382·37-s − 1.42·38-s + 0.996·40-s + 1.69·41-s + 0.340·43-s − 0.0181·44-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

\(L(2)\) \(\approx\) \(2.137626272\)
\(L(\frac12)\) \(\approx\) \(2.137626272\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.64T + 8T^{2} \)
5 \( 1 - 10.5T + 125T^{2} \)
7 \( 1 - 22T + 343T^{2} \)
11 \( 1 - 5.29T + 1.33e3T^{2} \)
17 \( 1 - 116.T + 4.91e3T^{2} \)
19 \( 1 - 126T + 6.85e3T^{2} \)
23 \( 1 - 31.7T + 1.21e4T^{2} \)
29 \( 1 - 52.9T + 2.43e4T^{2} \)
31 \( 1 - 182T + 2.97e4T^{2} \)
37 \( 1 - 86T + 5.06e4T^{2} \)
41 \( 1 - 444.T + 6.89e4T^{2} \)
43 \( 1 - 96T + 7.95e4T^{2} \)
47 \( 1 + 365.T + 1.03e5T^{2} \)
53 \( 1 + 190.T + 1.48e5T^{2} \)
59 \( 1 - 587.T + 2.05e5T^{2} \)
61 \( 1 - 574T + 2.26e5T^{2} \)
67 \( 1 - 530T + 3.00e5T^{2} \)
71 \( 1 + 809.T + 3.57e5T^{2} \)
73 \( 1 - 154T + 3.89e5T^{2} \)
79 \( 1 + 460T + 4.93e5T^{2} \)
83 \( 1 - 322.T + 5.71e5T^{2} \)
89 \( 1 + 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + 70T + 9.12e5T^{2} \)
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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

−9.316259147983626776363914704937, −8.132493870776209318655748984017, −7.918886333176190709910924481555, −6.92442745339876701374555524087, −5.62698257228675113089121181293, −5.16526231631242769386154676796, −4.11488930760148933405531250414, −2.72653866790398621540479129883, −1.44663337187210115042919382354, −0.983249373239673445682942248340, 0.983249373239673445682942248340, 1.44663337187210115042919382354, 2.72653866790398621540479129883, 4.11488930760148933405531250414, 5.16526231631242769386154676796, 5.62698257228675113089121181293, 6.92442745339876701374555524087, 7.918886333176190709910924481555, 8.132493870776209318655748984017, 9.316259147983626776363914704937

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