↑

Properties

Label 2-38808-1.1-c1-0-45
Degree $2$
Conductor $38808$
Sign $-1$
Analytic cond. $309.883$
Root an. cond. $17.6035$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 11-s + 3·13-s + 4·17-s + 7·19-s + 2·23-s + 4·25-s + 3·29-s − 11·37-s + 8·41-s − 2·43-s − 9·47-s − 10·53-s + 3·55-s − 5·59-s + 10·61-s − 9·65-s − 3·67-s − 8·71-s + 9·73-s − 4·79-s − 4·83-s − 12·85-s − 14·89-s − 21·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.301·11-s + 0.832·13-s + 0.970·17-s + 1.60·19-s + 0.417·23-s + 4/5·25-s + 0.557·29-s − 1.80·37-s + 1.24·41-s − 0.304·43-s − 1.31·47-s − 1.37·53-s + 0.404·55-s − 0.650·59-s + 1.28·61-s − 1.11·65-s − 0.366·67-s − 0.949·71-s + 1.05·73-s − 0.450·79-s − 0.439·83-s − 1.30·85-s − 1.48·89-s − 2.15·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38808\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(309.883\)
Root analytic conductor: \(17.6035\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38808,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.12843563018479, −14.61572517038296, −13.99159332725345, −13.66645994594572, −12.86542660073801, −12.39081625650645, −11.95628969556942, −11.37462297833678, −11.08993520577905, −10.38327664052075, −9.809152645201576, −9.237597993077843, −8.510631221755837, −8.077724498044843, −7.641399237597409, −7.087672003542990, −6.520211733623008, −5.632650977437700, −5.216965335594703, −4.509550530891319, −3.810080156757827, −3.253971926035005, −2.914381045877633, −1.587418248317392, −0.9682007898736402, 0, 0.9682007898736402, 1.587418248317392, 2.914381045877633, 3.253971926035005, 3.810080156757827, 4.509550530891319, 5.216965335594703, 5.632650977437700, 6.520211733623008, 7.087672003542990, 7.641399237597409, 8.077724498044843, 8.510631221755837, 9.237597993077843, 9.809152645201576, 10.38327664052075, 11.08993520577905, 11.37462297833678, 11.95628969556942, 12.39081625650645, 12.86542660073801, 13.66645994594572, 13.99159332725345, 14.61572517038296, 15.12843563018479

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