Properties

Label 2-39e2-1.1-c3-0-66
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  − 0.213·2-s − 7.95·4-s − 15.3·5-s − 32.3·7-s + 3.40·8-s + 3.27·10-s − 29.5·11-s + 6.92·14-s + 62.9·16-s + 78.1·17-s + 10.6·19-s + 122.·20-s + 6.32·22-s + 26.8·23-s + 110.·25-s + 257.·28-s + 190.·29-s − 128.·31-s − 40.7·32-s − 16.7·34-s + 496.·35-s − 379.·37-s − 2.27·38-s − 52.2·40-s + 464.·41-s + 322.·43-s + 235.·44-s + ⋯
L(s)  = 1  − 0.0755·2-s − 0.994·4-s − 1.37·5-s − 1.74·7-s + 0.150·8-s + 0.103·10-s − 0.811·11-s + 0.132·14-s + 0.982·16-s + 1.11·17-s + 0.128·19-s + 1.36·20-s + 0.0612·22-s + 0.243·23-s + 0.882·25-s + 1.73·28-s + 1.22·29-s − 0.742·31-s − 0.224·32-s − 0.0842·34-s + 2.39·35-s − 1.68·37-s − 0.00972·38-s − 0.206·40-s + 1.76·41-s + 1.14·43-s + 0.806·44-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 \( 1 + 0.213T + 8T^{2} \)
5 \( 1 + 15.3T + 125T^{2} \)
7 \( 1 + 32.3T + 343T^{2} \)
11 \( 1 + 29.5T + 1.33e3T^{2} \)
17 \( 1 - 78.1T + 4.91e3T^{2} \)
19 \( 1 - 10.6T + 6.85e3T^{2} \)
23 \( 1 - 26.8T + 1.21e4T^{2} \)
29 \( 1 - 190.T + 2.43e4T^{2} \)
31 \( 1 + 128.T + 2.97e4T^{2} \)
37 \( 1 + 379.T + 5.06e4T^{2} \)
41 \( 1 - 464.T + 6.89e4T^{2} \)
43 \( 1 - 322.T + 7.95e4T^{2} \)
47 \( 1 - 248.T + 1.03e5T^{2} \)
53 \( 1 + 740.T + 1.48e5T^{2} \)
59 \( 1 - 340.T + 2.05e5T^{2} \)
61 \( 1 + 590.T + 2.26e5T^{2} \)
67 \( 1 + 340.T + 3.00e5T^{2} \)
71 \( 1 - 36.2T + 3.57e5T^{2} \)
73 \( 1 + 164.T + 3.89e5T^{2} \)
79 \( 1 + 327.T + 4.93e5T^{2} \)
83 \( 1 - 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + 736.T + 7.04e5T^{2} \)
97 \( 1 - 1.49e3T + 9.12e5T^{2} \)
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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

−8.795854583549857802179743748192, −7.79363433016573897029552199956, −7.39001634755461943463090230930, −6.22118041551602546734409483264, −5.32758367173269904244581158372, −4.31671036116242977076909443109, −3.50472444676065134676349715726, −2.95902343843149850344263285633, −0.77148596944568927873868130035, 0, 0.77148596944568927873868130035, 2.95902343843149850344263285633, 3.50472444676065134676349715726, 4.31671036116242977076909443109, 5.32758367173269904244581158372, 6.22118041551602546734409483264, 7.39001634755461943463090230930, 7.79363433016573897029552199956, 8.795854583549857802179743748192

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