Properties

Label 2-39e2-1.1-c1-0-19
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  − 2·4-s + 5.19·7-s + 4·16-s + 3.46·19-s − 5·25-s − 10.3·28-s − 8.66·31-s + 6.92·37-s + 13·43-s + 20·49-s + 13·61-s − 8·64-s − 12.1·67-s + 1.73·73-s − 6.92·76-s + 13·79-s + 19.0·97-s + 10·100-s + 13·103-s + 8.66·109-s + 20.7·112-s + ⋯
L(s)  = 1  − 4-s + 1.96·7-s + 16-s + 0.794·19-s − 25-s − 1.96·28-s − 1.55·31-s + 1.13·37-s + 1.98·43-s + 2.85·49-s + 1.66·61-s − 64-s − 1.48·67-s + 0.202·73-s − 0.794·76-s + 1.46·79-s + 1.93·97-s + 100-s + 1.28·103-s + 0.829·109-s + 1.96·112-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: $\chi_{1521} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695954005\)
\(L(\frac12)\) \(\approx\) \(1.695954005\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 5.19T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.66T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−9.278550982516379096278423471452, −8.728135637863489422975384731820, −7.72536714583701354336606708079, −7.57777450618659838656119570441, −5.86759524499834832980391948758, −5.24169675575209242293511872037, −4.48531479108800978823334132249, −3.72835706033966006853332815739, −2.15113593829640636363996300451, −0.998216983262098798233120834662, 0.998216983262098798233120834662, 2.15113593829640636363996300451, 3.72835706033966006853332815739, 4.48531479108800978823334132249, 5.24169675575209242293511872037, 5.86759524499834832980391948758, 7.57777450618659838656119570441, 7.72536714583701354336606708079, 8.728135637863489422975384731820, 9.278550982516379096278423471452

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