Properties

Label 2-39e2-1.1-c3-0-78
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  − 2.36·2-s − 2.42·4-s − 6.42·5-s − 29.4·7-s + 24.6·8-s + 15.1·10-s + 0.624·11-s + 69.6·14-s − 38.7·16-s − 87.7·17-s + 82.8·19-s + 15.5·20-s − 1.47·22-s + 74.7·23-s − 83.7·25-s + 71.4·28-s − 226.·29-s + 173.·31-s − 105.·32-s + 207.·34-s + 189.·35-s + 112.·37-s − 195.·38-s − 158.·40-s + 267.·41-s + 383.·43-s − 1.51·44-s + ⋯
L(s)  = 1  − 0.835·2-s − 0.302·4-s − 0.574·5-s − 1.59·7-s + 1.08·8-s + 0.479·10-s + 0.0171·11-s + 1.32·14-s − 0.605·16-s − 1.25·17-s + 0.999·19-s + 0.173·20-s − 0.0142·22-s + 0.678·23-s − 0.670·25-s + 0.482·28-s − 1.44·29-s + 1.00·31-s − 0.582·32-s + 1.04·34-s + 0.914·35-s + 0.497·37-s − 0.834·38-s − 0.624·40-s + 1.01·41-s + 1.35·43-s − 0.00518·44-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 \( 1 + 2.36T + 8T^{2} \)
5 \( 1 + 6.42T + 125T^{2} \)
7 \( 1 + 29.4T + 343T^{2} \)
11 \( 1 - 0.624T + 1.33e3T^{2} \)
17 \( 1 + 87.7T + 4.91e3T^{2} \)
19 \( 1 - 82.8T + 6.85e3T^{2} \)
23 \( 1 - 74.7T + 1.21e4T^{2} \)
29 \( 1 + 226.T + 2.43e4T^{2} \)
31 \( 1 - 173.T + 2.97e4T^{2} \)
37 \( 1 - 112.T + 5.06e4T^{2} \)
41 \( 1 - 267.T + 6.89e4T^{2} \)
43 \( 1 - 383.T + 7.95e4T^{2} \)
47 \( 1 + 337.T + 1.03e5T^{2} \)
53 \( 1 - 146.T + 1.48e5T^{2} \)
59 \( 1 - 529.T + 2.05e5T^{2} \)
61 \( 1 - 203.T + 2.26e5T^{2} \)
67 \( 1 - 121.T + 3.00e5T^{2} \)
71 \( 1 - 661.T + 3.57e5T^{2} \)
73 \( 1 - 167.T + 3.89e5T^{2} \)
79 \( 1 + 101.T + 4.93e5T^{2} \)
83 \( 1 - 506.T + 5.71e5T^{2} \)
89 \( 1 + 1.40e3T + 7.04e5T^{2} \)
97 \( 1 - 1.90e3T + 9.12e5T^{2} \)
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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

−8.918628115732156826341584173651, −7.945143728784910784341585954333, −7.24870822471231370101241143971, −6.49952571990433946354298796322, −5.45190213179498770103211177960, −4.28423118064832192250339290215, −3.58939149680003314698650400328, −2.43724007709548561130663359666, −0.852543089929135927742108642246, 0, 0.852543089929135927742108642246, 2.43724007709548561130663359666, 3.58939149680003314698650400328, 4.28423118064832192250339290215, 5.45190213179498770103211177960, 6.49952571990433946354298796322, 7.24870822471231370101241143971, 7.945143728784910784341585954333, 8.918628115732156826341584173651

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