Properties

Label 2-39e2-1.1-c1-0-50
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.28·2-s − 0.356·4-s − 2.16·5-s + 1.04·7-s − 3.02·8-s − 2.78·10-s + 4.47·11-s + 1.34·14-s − 3.15·16-s − 1.67·17-s − 6.13·19-s + 0.774·20-s + 5.74·22-s + 6.78·23-s − 0.295·25-s − 0.374·28-s − 9.80·29-s − 7.91·31-s + 1.99·32-s − 2.14·34-s − 2.27·35-s − 10.1·37-s − 7.86·38-s + 6.55·40-s − 7.32·41-s − 0.506·43-s − 1.59·44-s + ⋯
L(s)  = 1  + 0.906·2-s − 0.178·4-s − 0.969·5-s + 0.396·7-s − 1.06·8-s − 0.879·10-s + 1.35·11-s + 0.359·14-s − 0.789·16-s − 0.406·17-s − 1.40·19-s + 0.173·20-s + 1.22·22-s + 1.41·23-s − 0.0591·25-s − 0.0707·28-s − 1.82·29-s − 1.42·31-s + 0.352·32-s − 0.368·34-s − 0.384·35-s − 1.66·37-s − 1.27·38-s + 1.03·40-s − 1.14·41-s − 0.0771·43-s − 0.240·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.28T + 2T^{2} \)
5 \( 1 + 2.16T + 5T^{2} \)
7 \( 1 - 1.04T + 7T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 + 6.13T + 19T^{2} \)
23 \( 1 - 6.78T + 23T^{2} \)
29 \( 1 + 9.80T + 29T^{2} \)
31 \( 1 + 7.91T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 7.32T + 41T^{2} \)
43 \( 1 + 0.506T + 43T^{2} \)
47 \( 1 + 0.887T + 47T^{2} \)
53 \( 1 + 5.11T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 1.77T + 67T^{2} \)
71 \( 1 - 1.36T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 - 3.87T + 79T^{2} \)
83 \( 1 + 4.95T + 83T^{2} \)
89 \( 1 + 4.81T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.823224933012562442036060225907, −8.515229802945749306592386321439, −7.22859199333072206252404204647, −6.63826547301163654180648468968, −5.55018319951028662796986349673, −4.73378026642536410377110283432, −3.87786765902530675949602612546, −3.48950618314885197331659682943, −1.84874921150038515210556530714, 0, 1.84874921150038515210556530714, 3.48950618314885197331659682943, 3.87786765902530675949602612546, 4.73378026642536410377110283432, 5.55018319951028662796986349673, 6.63826547301163654180648468968, 7.22859199333072206252404204647, 8.515229802945749306592386321439, 8.823224933012562442036060225907

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