↑

Properties

Label 2-29988-1.1-c1-0-14
Degree $2$
Conductor $29988$
Sign $1$
Analytic cond. $239.455$
Root an. cond. $15.4743$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 11-s − 3·13-s − 17-s + 2·19-s − 4·23-s + 11·25-s − 8·31-s + 8·37-s + 10·43-s − 10·47-s − 3·53-s − 4·55-s − 14·59-s + 8·61-s − 12·65-s − 10·67-s + 5·71-s + 16·73-s + 11·79-s + 12·83-s − 4·85-s + 9·89-s + 8·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.301·11-s − 0.832·13-s − 0.242·17-s + 0.458·19-s − 0.834·23-s + 11/5·25-s − 1.43·31-s + 1.31·37-s + 1.52·43-s − 1.45·47-s − 0.412·53-s − 0.539·55-s − 1.82·59-s + 1.02·61-s − 1.48·65-s − 1.22·67-s + 0.593·71-s + 1.87·73-s + 1.23·79-s + 1.31·83-s − 0.433·85-s + 0.953·89-s + 0.820·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29988\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(239.455\)
Root analytic conductor: \(15.4743\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29988,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.025627572\)
\(L(\frac12)\) \(\approx\) \(3.025627572\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−14.97906750049292, −14.45797478572740, −14.11486558550200, −13.57764667781177, −13.04986299308015, −12.61935037757258, −12.11297272279606, −11.23102842626176, −10.79337440609267, −10.21444372384678, −9.590211435989144, −9.416678510707384, −8.845538785905183, −7.842397063368304, −7.573718520909425, −6.621550420889357, −6.239542370782806, −5.633631858226494, −5.106012749006491, −4.575727799891536, −3.595112932860661, −2.782291447980040, −2.173551089023896, −1.707625068706323, −0.6456610334919439, 0.6456610334919439, 1.707625068706323, 2.173551089023896, 2.782291447980040, 3.595112932860661, 4.575727799891536, 5.106012749006491, 5.633631858226494, 6.239542370782806, 6.621550420889357, 7.573718520909425, 7.842397063368304, 8.845538785905183, 9.416678510707384, 9.590211435989144, 10.21444372384678, 10.79337440609267, 11.23102842626176, 12.11297272279606, 12.61935037757258, 13.04986299308015, 13.57764667781177, 14.11486558550200, 14.45797478572740, 14.97906750049292

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