Properties

Label 2-143e2-1.1-c1-0-5
Degree $2$
Conductor $20449$
Sign $1$
Analytic cond. $163.286$
Root an. cond. $12.7783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 2·9-s + 2·10-s − 2·12-s + 4·14-s + 15-s − 4·16-s + 2·17-s + 4·18-s − 2·20-s + 2·21-s − 23-s − 4·25-s + 5·27-s − 4·28-s − 2·30-s − 7·31-s + 8·32-s − 4·34-s + 2·35-s − 4·36-s − 3·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.577·12-s + 1.06·14-s + 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s − 0.447·20-s + 0.436·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s + 1.41·32-s − 0.685·34-s + 0.338·35-s − 2/3·36-s − 0.493·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20449\)    =    \(11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(163.286\)
Root analytic conductor: \(12.7783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 20449,\ (\ :1/2),\ 1)\)

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)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−16.33876970759384, −15.94136401912385, −15.36079251592833, −14.69187686722283, −13.92654662376284, −13.55158006948452, −12.55692172152170, −12.33332213238782, −11.48257315472483, −11.17381612847762, −10.61077523102692, −9.997803318826309, −9.470201127780350, −9.047361338686055, −8.217506491730922, −7.972000332755686, −7.196250039602744, −6.644997946132623, −6.035780196715335, −5.348640601367650, −4.615561361841008, −3.663131710551717, −3.082857453782681, −2.079764001679285, −1.244472589701135, 0, 0, 1.244472589701135, 2.079764001679285, 3.082857453782681, 3.663131710551717, 4.615561361841008, 5.348640601367650, 6.035780196715335, 6.644997946132623, 7.196250039602744, 7.972000332755686, 8.217506491730922, 9.047361338686055, 9.470201127780350, 9.997803318826309, 10.61077523102692, 11.17381612847762, 11.48257315472483, 12.33332213238782, 12.55692172152170, 13.55158006948452, 13.92654662376284, 14.69187686722283, 15.36079251592833, 15.94136401912385, 16.33876970759384

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