Properties

Label 2-39e2-1.1-c1-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.628·2-s − 1.60·4-s + 4.14·5-s + 2.26·8-s − 2.60·10-s + 5.40·11-s + 1.78·16-s − 6.66·20-s − 3.39·22-s + 12.2·25-s − 5.65·32-s + 9.39·40-s − 1.63·41-s + 4·43-s − 8.67·44-s − 13.7·47-s − 7·49-s − 7.66·50-s + 22.4·55-s + 11.1·59-s − 7.21·61-s − 0.0277·64-s + 7.91·71-s + 14.4·79-s + 7.42·80-s + 1.02·82-s + 0.380·83-s + ⋯
L(s)  = 1  − 0.444·2-s − 0.802·4-s + 1.85·5-s + 0.800·8-s − 0.823·10-s + 1.62·11-s + 0.447·16-s − 1.48·20-s − 0.723·22-s + 2.44·25-s − 0.999·32-s + 1.48·40-s − 0.255·41-s + 0.609·43-s − 1.30·44-s − 1.99·47-s − 49-s − 1.08·50-s + 3.02·55-s + 1.45·59-s − 0.923·61-s − 0.00346·64-s + 0.939·71-s + 1.62·79-s + 0.829·80-s + 0.113·82-s + 0.0417·83-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: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.783428205\)
\(L(\frac12)\) \(\approx\) \(1.783428205\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.628T + 2T^{2} \)
5 \( 1 - 4.14T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 5.40T + 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 1.63T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 13.7T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 7.91T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 0.380T + 83T^{2} \)
89 \( 1 + 9.93T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.467413541011298188758191244593, −8.971839167611558545694176763821, −8.150847066190150547048612623157, −6.85719035145052777952570035421, −6.26407300078948751418197268499, −5.37516921414065477244529195254, −4.56456954118016983866394286818, −3.42785021766152012935178947781, −1.96680969462411954523886484275, −1.14339478440729767047013499426, 1.14339478440729767047013499426, 1.96680969462411954523886484275, 3.42785021766152012935178947781, 4.56456954118016983866394286818, 5.37516921414065477244529195254, 6.26407300078948751418197268499, 6.85719035145052777952570035421, 8.150847066190150547048612623157, 8.971839167611558545694176763821, 9.467413541011298188758191244593

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