Properties

Label 2-39e2-1.1-c1-0-38
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  − 1.73·2-s + 0.999·4-s + 3.46·7-s + 1.73·8-s − 3.46·11-s − 5.99·14-s − 5·16-s − 6·17-s + 3.46·19-s + 5.99·22-s − 5·25-s + 3.46·28-s − 6·29-s − 3.46·31-s + 5.19·32-s + 10.3·34-s − 6.92·37-s − 5.99·38-s + 6.92·41-s + 4·43-s − 3.46·44-s + 3.46·47-s + 4.99·49-s + 8.66·50-s − 6·53-s + 6.00·56-s + 10.3·58-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s + 1.30·7-s + 0.612·8-s − 1.04·11-s − 1.60·14-s − 1.25·16-s − 1.45·17-s + 0.794·19-s + 1.27·22-s − 25-s + 0.654·28-s − 1.11·29-s − 0.622·31-s + 0.918·32-s + 1.78·34-s − 1.13·37-s − 0.973·38-s + 1.08·41-s + 0.609·43-s − 0.522·44-s + 0.505·47-s + 0.714·49-s + 1.22·50-s − 0.824·53-s + 0.801·56-s + 1.36·58-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: \(1\)
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 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−8.998764442342834100165097184971, −8.332257825284256696630759004412, −7.61922832362562448685817427313, −7.17482099423046659262209518890, −5.72983536965446822808509467013, −4.92051569999899627639582107947, −4.06264551255806293159961850586, −2.39455644807448585223645387954, −1.55463176831811738397772128768, 0, 1.55463176831811738397772128768, 2.39455644807448585223645387954, 4.06264551255806293159961850586, 4.92051569999899627639582107947, 5.72983536965446822808509467013, 7.17482099423046659262209518890, 7.61922832362562448685817427313, 8.332257825284256696630759004412, 8.998764442342834100165097184971

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