↑

Properties

Label 2-2720-136.101-c1-0-9
Degree $2$
Conductor $2720$
Sign $-0.140 - 0.990i$
Analytic cond. $21.7193$
Root an. cond. $4.66039$
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  + 0.528·3-s + 5-s − 4.36i·7-s − 2.72·9-s − 0.873·11-s + 6.88i·13-s + 0.528·15-s + (−3.39 + 2.33i)17-s + 3.88i·19-s − 2.30i·21-s − 4.09i·23-s + 25-s − 3.02·27-s − 6.41·29-s + 6.12i·31-s + ⋯
L(s)  = 1  + 0.304·3-s + 0.447·5-s − 1.65i·7-s − 0.907·9-s − 0.263·11-s + 1.90i·13-s + 0.136·15-s + (−0.824 + 0.566i)17-s + 0.892i·19-s − 0.503i·21-s − 0.853i·23-s + 0.200·25-s − 0.581·27-s − 1.19·29-s + 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $-0.140 - 0.990i$
Analytic conductor: \(21.7193\)
Root analytic conductor: \(4.66039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (2481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :1/2),\ -0.140 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.068397735\)
\(L(\frac12)\) \(\approx\) \(1.068397735\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
17 \( 1 + (3.39 - 2.33i)T \)
good3 \( 1 - 0.528T + 3T^{2} \)
7 \( 1 + 4.36iT - 7T^{2} \)
11 \( 1 + 0.873T + 11T^{2} \)
13 \( 1 - 6.88iT - 13T^{2} \)
19 \( 1 - 3.88iT - 19T^{2} \)
23 \( 1 + 4.09iT - 23T^{2} \)
29 \( 1 + 6.41T + 29T^{2} \)
31 \( 1 - 6.12iT - 31T^{2} \)
37 \( 1 - 6.96T + 37T^{2} \)
41 \( 1 - 12.0iT - 41T^{2} \)
43 \( 1 - 9.29iT - 43T^{2} \)
47 \( 1 + 1.39T + 47T^{2} \)
53 \( 1 + 5.12iT - 53T^{2} \)
59 \( 1 + 5.18iT - 59T^{2} \)
61 \( 1 - 7.45T + 61T^{2} \)
67 \( 1 - 1.10iT - 67T^{2} \)
71 \( 1 + 8.33iT - 71T^{2} \)
73 \( 1 - 1.86iT - 73T^{2} \)
79 \( 1 - 3.18iT - 79T^{2} \)
83 \( 1 - 4.38iT - 83T^{2} \)
89 \( 1 - 9.98T + 89T^{2} \)
97 \( 1 - 6.33iT - 97T^{2} \)
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

−9.090355051748709872418687715343, −8.232243198963705112784391800907, −7.61032289583663799316539380124, −6.48465943701504834448036571334, −6.40507620851811885107758628636, −4.96070345602790222814062352697, −4.23738873612037043268313385908, −3.52405365611274766952255342911, −2.30767382456882737835192719155, −1.35756397871374475377613993814, 0.31451490801237355098367853657, 2.29421204247195466185900931226, 2.58482992000734125568503340041, 3.54329719126651233448452451732, 5.03732285172630100452204122177, 5.67904026842703198652985290384, 5.89550731468194404744396403602, 7.25362951933437926370695211860, 7.975454012490179443593529768130, 8.863636808169771039269788430945

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