Properties

Label 2-2e6-1.1-c13-0-4
Degree $2$
Conductor $64$
Sign $1$
Analytic cond. $68.6277$
Root an. cond. $8.28418$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.76e4·5-s − 1.59e6·9-s − 7.25e6·13-s − 8.58e6·17-s − 9.08e8·25-s + 1.48e9·29-s + 2.61e10·37-s − 4.83e10·41-s + 2.81e10·45-s − 9.68e10·49-s + 2.86e11·53-s + 3.08e11·61-s + 1.28e11·65-s + 2.58e12·73-s + 2.54e12·81-s + 1.51e11·85-s + 7.78e12·89-s + 1.08e13·97-s + 2.05e13·101-s + 2.11e13·109-s − 3.53e12·113-s + 1.15e13·117-s + ⋯
L(s)  = 1  − 0.505·5-s − 9-s − 0.416·13-s − 0.0862·17-s − 0.743·25-s + 0.464·29-s + 1.67·37-s − 1.58·41-s + 0.505·45-s − 49-s + 1.77·53-s + 0.766·61-s + 0.210·65-s + 1.99·73-s + 81-s + 0.0436·85-s + 1.65·89-s + 1.32·97-s + 1.92·101-s + 1.20·109-s − 0.159·113-s + 0.416·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(68.6277\)
Root analytic conductor: \(8.28418\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.272959675\)
\(L(\frac12)\) \(\approx\) \(1.272959675\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + p^{13} T^{2} \)
5 \( 1 + 17678 T + p^{13} T^{2} \)
7 \( 1 + p^{13} T^{2} \)
11 \( 1 + p^{13} T^{2} \)
13 \( 1 + 7254006 T + p^{13} T^{2} \)
17 \( 1 + 8582078 T + p^{13} T^{2} \)
19 \( 1 + p^{13} T^{2} \)
23 \( 1 + p^{13} T^{2} \)
29 \( 1 - 1486672730 T + p^{13} T^{2} \)
31 \( 1 + p^{13} T^{2} \)
37 \( 1 - 26174743922 T + p^{13} T^{2} \)
41 \( 1 + 48310539670 T + p^{13} T^{2} \)
43 \( 1 + p^{13} T^{2} \)
47 \( 1 + p^{13} T^{2} \)
53 \( 1 - 286798198946 T + p^{13} T^{2} \)
59 \( 1 + p^{13} T^{2} \)
61 \( 1 - 308331475930 T + p^{13} T^{2} \)
67 \( 1 + p^{13} T^{2} \)
71 \( 1 + p^{13} T^{2} \)
73 \( 1 - 2582975771994 T + p^{13} T^{2} \)
79 \( 1 + p^{13} T^{2} \)
83 \( 1 + p^{13} T^{2} \)
89 \( 1 - 7780304165930 T + p^{13} T^{2} \)
97 \( 1 - 10874684920338 T + p^{13} T^{2} \)
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

−12.01450317146701149363837327357, −11.28370655925478330737102044733, −9.952816548196404857177916642762, −8.667068551364035388868341314754, −7.68065145391708285263636049957, −6.28777913814001257874068942568, −4.99450360259779746057580373494, −3.59295171318543287746668332850, −2.31839339534315798404063901230, −0.57077185902908256993431499083, 0.57077185902908256993431499083, 2.31839339534315798404063901230, 3.59295171318543287746668332850, 4.99450360259779746057580373494, 6.28777913814001257874068942568, 7.68065145391708285263636049957, 8.667068551364035388868341314754, 9.952816548196404857177916642762, 11.28370655925478330737102044733, 12.01450317146701149363837327357

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