↑

Properties

Label 2-728-56.27-c1-0-81
Degree $2$
Conductor $728$
Sign $0.517 + 0.855i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 0.0740i)2-s − 1.48i·3-s + (1.98 + 0.209i)4-s + 1.02·5-s + (0.110 − 2.10i)6-s + (−1.70 − 2.02i)7-s + (2.79 + 0.442i)8-s + 0.784·9-s + (1.45 + 0.0763i)10-s + 0.608·11-s + (0.311 − 2.96i)12-s − 13-s + (−2.26 − 2.98i)14-s − 1.53i·15-s + (3.91 + 0.832i)16-s + 0.526i·17-s + ⋯
L(s)  = 1  + (0.998 + 0.0523i)2-s − 0.859i·3-s + (0.994 + 0.104i)4-s + 0.460·5-s + (0.0450 − 0.858i)6-s + (−0.645 − 0.763i)7-s + (0.987 + 0.156i)8-s + 0.261·9-s + (0.459 + 0.0241i)10-s + 0.183·11-s + (0.0899 − 0.854i)12-s − 0.277·13-s + (−0.604 − 0.796i)14-s − 0.395i·15-s + (0.978 + 0.208i)16-s + 0.127i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.517 + 0.855i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.517 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63851 - 1.48698i\)
\(L(\frac12)\) \(\approx\) \(2.63851 - 1.48698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0740i)T \)
7 \( 1 + (1.70 + 2.02i)T \)
13 \( 1 + T \)
good3 \( 1 + 1.48iT - 3T^{2} \)
5 \( 1 - 1.02T + 5T^{2} \)
11 \( 1 - 0.608T + 11T^{2} \)
17 \( 1 - 0.526iT - 17T^{2} \)
19 \( 1 + 1.94iT - 19T^{2} \)
23 \( 1 - 0.0153iT - 23T^{2} \)
29 \( 1 - 4.64iT - 29T^{2} \)
31 \( 1 - 4.57T + 31T^{2} \)
37 \( 1 + 2.15iT - 37T^{2} \)
41 \( 1 - 6.24iT - 41T^{2} \)
43 \( 1 + 3.29T + 43T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 + 1.16iT - 53T^{2} \)
59 \( 1 + 0.783iT - 59T^{2} \)
61 \( 1 - 5.32T + 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 - 9.18iT - 73T^{2} \)
79 \( 1 + 9.29iT - 79T^{2} \)
83 \( 1 + 1.45iT - 83T^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 - 8.46iT - 97T^{2} \)
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

−10.30116288434170545484887598352, −9.669606793924034247661581634622, −8.192540226629701164765320192136, −7.24863270970450956103103674509, −6.71872902895857476194579528678, −5.96581353679966549641955066571, −4.76545462855710667816546512867, −3.75112614069955890854325783533, −2.54872479031637143285409250681, −1.31238685977956573321186544040, 1.99130726445158891985733225352, 3.16262844820513198298069935910, 4.11266791122236518910339747175, 5.06651502761658901651505110651, 5.90727040825319046266226247963, 6.67046019093586352570430757824, 7.82330486506418603250980483321, 9.108264162447474051625472360491, 9.926494415781171894648421775051, 10.36628782321115462553719447554

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