↑

Properties

Label 2-2e5-8.3-c20-0-8
Degree $2$
Conductor $32$
Sign $0.956 + 0.290i$
Analytic cond. $81.1244$
Root an. cond. $9.00690$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.18e4·3-s − 4.86e6i·5-s + 3.96e8i·7-s − 3.34e9·9-s − 2.78e10·11-s − 1.92e11i·13-s − 5.75e10i·15-s + 1.00e12·17-s − 9.04e11·19-s + 4.69e12i·21-s + 5.92e13i·23-s + 7.17e13·25-s − 8.08e13·27-s − 4.75e14i·29-s − 5.26e14i·31-s + ⋯
L(s)  = 1  + 0.200·3-s − 0.497i·5-s + 1.40i·7-s − 0.959·9-s − 1.07·11-s − 1.39i·13-s − 0.0997i·15-s + 0.499·17-s − 0.147·19-s + 0.281i·21-s + 1.43i·23-s + 0.752·25-s − 0.392·27-s − 1.13i·29-s − 0.642i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(81.1244\)
Root analytic conductor: \(9.00690\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :10),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(1.658201353\)
\(L(\frac12)\) \(\approx\) \(1.658201353\)
\(L(11)\) 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 \( 1 - 1.18e4T + 3.48e9T^{2} \)
5 \( 1 + 4.86e6iT - 9.53e13T^{2} \)
7 \( 1 - 3.96e8iT - 7.97e16T^{2} \)
11 \( 1 + 2.78e10T + 6.72e20T^{2} \)
13 \( 1 + 1.92e11iT - 1.90e22T^{2} \)
17 \( 1 - 1.00e12T + 4.06e24T^{2} \)
19 \( 1 + 9.04e11T + 3.75e25T^{2} \)
23 \( 1 - 5.92e13iT - 1.71e27T^{2} \)
29 \( 1 + 4.75e14iT - 1.76e29T^{2} \)
31 \( 1 + 5.26e14iT - 6.71e29T^{2} \)
37 \( 1 - 5.41e15iT - 2.31e31T^{2} \)
41 \( 1 + 1.00e16T + 1.80e32T^{2} \)
43 \( 1 - 2.73e16T + 4.67e32T^{2} \)
47 \( 1 + 5.52e16iT - 2.76e33T^{2} \)
53 \( 1 - 1.63e16iT - 3.05e34T^{2} \)
59 \( 1 - 5.08e17T + 2.61e35T^{2} \)
61 \( 1 - 2.76e17iT - 5.08e35T^{2} \)
67 \( 1 - 7.08e17T + 3.32e36T^{2} \)
71 \( 1 - 2.20e18iT - 1.05e37T^{2} \)
73 \( 1 - 8.31e18T + 1.84e37T^{2} \)
79 \( 1 - 2.46e18iT - 8.96e37T^{2} \)
83 \( 1 + 1.22e19T + 2.40e38T^{2} \)
89 \( 1 - 3.83e19T + 9.72e38T^{2} \)
97 \( 1 - 2.89e19T + 5.43e39T^{2} \)
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

−12.48467368844345169371280206365, −11.40726629464995345833628705429, −9.907158309931593144127033568384, −8.625561130217868944218658865324, −7.894369771290735589212828365311, −5.74826761829536639926680729095, −5.26027260843964284305864516606, −3.17583499980166742110432720309, −2.30690992182170878075307129057, −0.55659171758142293589839776326, 0.70133532144315317949052334380, 2.33648515094496038076503609897, 3.54151176503162002228039311616, 4.86519319318517925213423149700, 6.52816517205750551967035677909, 7.54064398065291423505409288166, 8.850047483123769207572180848743, 10.40326980547330115215178122059, 11.07615992018541185036572393165, 12.66086814860644150009535177164

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