Properties

Label 2-39e2-1.1-c3-0-177
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  + 4.69·2-s + 14.0·4-s + 4.47·5-s − 27.2·7-s + 28.5·8-s + 21.0·10-s − 5.99·11-s − 127.·14-s + 21.4·16-s − 105.·17-s + 156.·19-s + 62.9·20-s − 28.1·22-s + 175.·23-s − 104.·25-s − 382.·28-s − 204.·29-s − 31.9·31-s − 127.·32-s − 493.·34-s − 121.·35-s − 344.·37-s + 735.·38-s + 127.·40-s + 46.5·41-s − 173.·43-s − 84.3·44-s + ⋯
L(s)  = 1  + 1.66·2-s + 1.75·4-s + 0.400·5-s − 1.46·7-s + 1.26·8-s + 0.664·10-s − 0.164·11-s − 2.44·14-s + 0.335·16-s − 1.49·17-s + 1.88·19-s + 0.703·20-s − 0.272·22-s + 1.59·23-s − 0.839·25-s − 2.58·28-s − 1.31·29-s − 0.185·31-s − 0.703·32-s − 2.48·34-s − 0.587·35-s − 1.52·37-s + 3.13·38-s + 0.504·40-s + 0.177·41-s − 0.614·43-s − 0.288·44-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 \( 1 - 4.69T + 8T^{2} \)
5 \( 1 - 4.47T + 125T^{2} \)
7 \( 1 + 27.2T + 343T^{2} \)
11 \( 1 + 5.99T + 1.33e3T^{2} \)
17 \( 1 + 105.T + 4.91e3T^{2} \)
19 \( 1 - 156.T + 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
29 \( 1 + 204.T + 2.43e4T^{2} \)
31 \( 1 + 31.9T + 2.97e4T^{2} \)
37 \( 1 + 344.T + 5.06e4T^{2} \)
41 \( 1 - 46.5T + 6.89e4T^{2} \)
43 \( 1 + 173.T + 7.95e4T^{2} \)
47 \( 1 + 265.T + 1.03e5T^{2} \)
53 \( 1 + 172.T + 1.48e5T^{2} \)
59 \( 1 + 137.T + 2.05e5T^{2} \)
61 \( 1 + 58.9T + 2.26e5T^{2} \)
67 \( 1 + 211.T + 3.00e5T^{2} \)
71 \( 1 + 436.T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3T + 3.89e5T^{2} \)
79 \( 1 + 1.01e3T + 4.93e5T^{2} \)
83 \( 1 + 150.T + 5.71e5T^{2} \)
89 \( 1 - 565.T + 7.04e5T^{2} \)
97 \( 1 - 286.T + 9.12e5T^{2} \)
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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

−8.951807150797317708859091043703, −7.30529383855687195654438332828, −6.88747080498973190164394673025, −6.00523801314290213026860058468, −5.40643369036703236838257455788, −4.52943315625764752602754256770, −3.38233036020837022782639455566, −3.03929746714142521518696885274, −1.82351625270297335094613494892, 0, 1.82351625270297335094613494892, 3.03929746714142521518696885274, 3.38233036020837022782639455566, 4.52943315625764752602754256770, 5.40643369036703236838257455788, 6.00523801314290213026860058468, 6.88747080498973190164394673025, 7.30529383855687195654438332828, 8.951807150797317708859091043703

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