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Properties

Label 2-2e5-8.3-c20-0-18
Degree $2$
Conductor $32$
Sign $-0.957 + 0.287i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.63e4·3-s − 1.68e7i·5-s − 3.64e8i·7-s + 3.97e9·9-s − 2.95e10·11-s − 1.06e11i·13-s − 1.45e12i·15-s + 2.31e12·17-s + 8.42e11·19-s − 3.15e13i·21-s + 2.92e12i·23-s − 1.87e14·25-s + 4.23e13·27-s − 2.89e14i·29-s + 6.94e14i·31-s + ⋯
L(s)  = 1  + 1.46·3-s − 1.72i·5-s − 1.29i·7-s + 1.14·9-s − 1.13·11-s − 0.772i·13-s − 2.51i·15-s + 1.14·17-s + 0.137·19-s − 1.88i·21-s + 0.0706i·23-s − 1.96·25-s + 0.205·27-s − 0.687i·29-s + 0.847i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.957 + 0.287i$
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: \((2,\ 32,\ (\ :10),\ -0.957 + 0.287i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(2.904388467\)
\(L(\frac12)\) \(\approx\) \(2.904388467\)
\(L(11)\) 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 \( 1 - 8.63e4T + 3.48e9T^{2} \)
5 \( 1 + 1.68e7iT - 9.53e13T^{2} \)
7 \( 1 + 3.64e8iT - 7.97e16T^{2} \)
11 \( 1 + 2.95e10T + 6.72e20T^{2} \)
13 \( 1 + 1.06e11iT - 1.90e22T^{2} \)
17 \( 1 - 2.31e12T + 4.06e24T^{2} \)
19 \( 1 - 8.42e11T + 3.75e25T^{2} \)
23 \( 1 - 2.92e12iT - 1.71e27T^{2} \)
29 \( 1 + 2.89e14iT - 1.76e29T^{2} \)
31 \( 1 - 6.94e14iT - 6.71e29T^{2} \)
37 \( 1 - 2.39e15iT - 2.31e31T^{2} \)
41 \( 1 + 8.44e15T + 1.80e32T^{2} \)
43 \( 1 - 2.60e16T + 4.67e32T^{2} \)
47 \( 1 - 4.58e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.23e17iT - 3.05e34T^{2} \)
59 \( 1 - 3.72e17T + 2.61e35T^{2} \)
61 \( 1 + 1.14e18iT - 5.08e35T^{2} \)
67 \( 1 + 1.69e18T + 3.32e36T^{2} \)
71 \( 1 + 1.43e18iT - 1.05e37T^{2} \)
73 \( 1 - 6.55e17T + 1.84e37T^{2} \)
79 \( 1 - 3.74e18iT - 8.96e37T^{2} \)
83 \( 1 - 1.51e19T + 2.40e38T^{2} \)
89 \( 1 - 2.12e19T + 9.72e38T^{2} \)
97 \( 1 + 6.94e19T + 5.43e39T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.46644899437374139023669945699, −10.37611222099682957806501067739, −9.358680290000983008886870779779, −8.108281515130575452252452729165, −7.70424887171089255856701680186, −5.32397567408717171160110791157, −4.16550858571494699388810715292, −2.99795621917416167924425849053, −1.43731504059966198208676352670, −0.51255607591547275695486927524, 2.13472317790026588542271236669, 2.68026169345001335126066620602, 3.57350461964812419603332936019, 5.66783654055019537951487962605, 7.14729634063330000867543308257, 8.124434735479692805252802799819, 9.351747670213761985754824346156, 10.44237966434819512007350211041, 11.84263352715054118509568403899, 13.38491563928609893999001571068

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