Properties

Label 2-159936-1.1-c1-0-127
Degree $2$
Conductor $159936$
Sign $1$
Analytic cond. $1277.09$
Root an. cond. $35.7364$
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  + 3-s − 3·5-s + 9-s + 6·11-s + 5·13-s − 3·15-s + 17-s + 2·19-s + 6·23-s + 4·25-s + 27-s + 9·29-s + 7·31-s + 6·33-s − 2·37-s + 5·39-s − 9·41-s − 8·43-s − 3·45-s − 3·47-s + 51-s − 18·55-s + 2·57-s + 3·59-s + 8·61-s − 15·65-s − 8·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1/3·9-s + 1.80·11-s + 1.38·13-s − 0.774·15-s + 0.242·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 0.192·27-s + 1.67·29-s + 1.25·31-s + 1.04·33-s − 0.328·37-s + 0.800·39-s − 1.40·41-s − 1.21·43-s − 0.447·45-s − 0.437·47-s + 0.140·51-s − 2.42·55-s + 0.264·57-s + 0.390·59-s + 1.02·61-s − 1.86·65-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159936\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1277.09\)
Root analytic conductor: \(35.7364\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159936} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.090814803\)
\(L(\frac12)\) \(\approx\) \(4.090814803\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.52927287806964, −12.71265676464736, −12.24289078867796, −11.78431333931454, −11.40728436447379, −11.21739713934300, −10.24458344272312, −10.05981320670746, −9.297242792182718, −8.716788665016633, −8.499038904683950, −8.185258536466335, −7.451672392671810, −6.879871420019442, −6.574625723295369, −6.157212614593343, −5.116302446131379, −4.743151584395682, −4.027347818798033, −3.713682329375892, −3.259651437087334, −2.784975428913541, −1.643994984079708, −1.169923833950968, −0.6653038152104891, 0.6653038152104891, 1.169923833950968, 1.643994984079708, 2.784975428913541, 3.259651437087334, 3.713682329375892, 4.027347818798033, 4.743151584395682, 5.116302446131379, 6.157212614593343, 6.574625723295369, 6.879871420019442, 7.451672392671810, 8.185258536466335, 8.499038904683950, 8.716788665016633, 9.297242792182718, 10.05981320670746, 10.24458344272312, 11.21739713934300, 11.40728436447379, 11.78431333931454, 12.24289078867796, 12.71265676464736, 13.52927287806964

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