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Properties

Label 2-2e5-32.3-c2-0-3
Degree $2$
Conductor $32$
Sign $0.968 - 0.250i$
Analytic cond. $0.871936$
Root an. cond. $0.933775$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.93 − 0.488i)2-s + (−1.73 + 4.18i)3-s + (3.52 − 1.89i)4-s + (−1.85 − 4.48i)5-s + (−1.31 + 8.95i)6-s + (−5.27 − 5.27i)7-s + (5.90 − 5.40i)8-s + (−8.12 − 8.12i)9-s + (−5.79 − 7.79i)10-s + (6.20 + 14.9i)11-s + (1.83 + 18.0i)12-s + (−4.22 + 10.2i)13-s + (−12.8 − 7.65i)14-s + 21.9·15-s + (8.80 − 13.3i)16-s − 2.84i·17-s + ⋯
L(s)  = 1  + (0.969 − 0.244i)2-s + (−0.577 + 1.39i)3-s + (0.880 − 0.474i)4-s + (−0.371 − 0.897i)5-s + (−0.219 + 1.49i)6-s + (−0.753 − 0.753i)7-s + (0.737 − 0.675i)8-s + (−0.902 − 0.902i)9-s + (−0.579 − 0.779i)10-s + (0.563 + 1.36i)11-s + (0.152 + 1.50i)12-s + (−0.325 + 0.784i)13-s + (−0.915 − 0.546i)14-s + 1.46·15-s + (0.550 − 0.834i)16-s − 0.167i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ 0.968 - 0.250i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30398 + 0.165971i\)
\(L(\frac12)\) \(\approx\) \(1.30398 + 0.165971i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.93 + 0.488i)T \)
good3 \( 1 + (1.73 - 4.18i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (1.85 + 4.48i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (5.27 + 5.27i)T + 49iT^{2} \)
11 \( 1 + (-6.20 - 14.9i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (4.22 - 10.2i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 2.84iT - 289T^{2} \)
19 \( 1 + (12.4 + 5.14i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (1.43 - 1.43i)T - 529iT^{2} \)
29 \( 1 + (-36.9 - 15.3i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 4.73iT - 961T^{2} \)
37 \( 1 + (6.68 + 16.1i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-40.4 - 40.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (24.5 + 59.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 16.5T + 2.20e3T^{2} \)
53 \( 1 + (46.9 - 19.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-50.0 + 20.7i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (54.3 + 22.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (25.5 - 61.5i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-7.12 - 7.12i)T + 5.04e3iT^{2} \)
73 \( 1 + (-55.3 - 55.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 11.0T + 6.24e3T^{2} \)
83 \( 1 + (29.9 + 12.4i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-16.7 + 16.7i)T - 7.92e3iT^{2} \)
97 \( 1 + 67.8T + 9.40e3T^{2} \)
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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

−16.37735938263667260884604902846, −15.58511238553685288162820493911, −14.40397230103404879010902384631, −12.79378224638134043019791583074, −11.80883829804259030013259794286, −10.42638375203472872289923041474, −9.462739280814463915452774583821, −6.77072661608046261871040648004, −4.80261727251954309821417842497, −4.09252982227681704744642463955, 2.99751965505994850454428959763, 5.95590517168597440508303564640, 6.63877806374874614630931626745, 8.075984692359006529629686489289, 10.91553012311301204854814673038, 11.97914868538614985454349741770, 12.81867187007108193671005344425, 13.95597821479136375099809687727, 15.18753595334088881998365562892, 16.45146502143166030641801814383

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