Properties

Label 2-2e5-32.27-c2-0-3
Degree $2$
Conductor $32$
Sign $0.757 - 0.652i$
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  + (−0.108 + 1.99i)2-s + (4.35 − 1.80i)3-s + (−3.97 − 0.432i)4-s + (−2.81 − 1.16i)5-s + (3.12 + 8.88i)6-s + (−6.23 + 6.23i)7-s + (1.29 − 7.89i)8-s + (9.32 − 9.32i)9-s + (2.63 − 5.49i)10-s + (−8.06 − 3.33i)11-s + (−18.0 + 5.28i)12-s + (13.3 − 5.51i)13-s + (−11.7 − 13.1i)14-s − 14.3·15-s + (15.6 + 3.43i)16-s − 4.56i·17-s + ⋯
L(s)  = 1  + (−0.0540 + 0.998i)2-s + (1.45 − 0.600i)3-s + (−0.994 − 0.108i)4-s + (−0.563 − 0.233i)5-s + (0.521 + 1.48i)6-s + (−0.890 + 0.890i)7-s + (0.161 − 0.986i)8-s + (1.03 − 1.03i)9-s + (0.263 − 0.549i)10-s + (−0.732 − 0.303i)11-s + (−1.50 + 0.440i)12-s + (1.02 − 0.424i)13-s + (−0.841 − 0.937i)14-s − 0.957·15-s + (0.976 + 0.214i)16-s − 0.268i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ 0.757 - 0.652i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08581 + 0.403218i\)
\(L(\frac12)\) \(\approx\) \(1.08581 + 0.403218i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.108 - 1.99i)T \)
good3 \( 1 + (-4.35 + 1.80i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (2.81 + 1.16i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (6.23 - 6.23i)T - 49iT^{2} \)
11 \( 1 + (8.06 + 3.33i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (-13.3 + 5.51i)T + (119. - 119. i)T^{2} \)
17 \( 1 + 4.56iT - 289T^{2} \)
19 \( 1 + (-13.4 - 32.4i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (6.75 + 6.75i)T + 529iT^{2} \)
29 \( 1 + (0.266 + 0.643i)T + (-594. + 594. i)T^{2} \)
31 \( 1 + 0.326iT - 961T^{2} \)
37 \( 1 + (-31.5 - 13.0i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (-15.7 + 15.7i)T - 1.68e3iT^{2} \)
43 \( 1 + (-4.83 - 2.00i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 49.7T + 2.20e3T^{2} \)
53 \( 1 + (-4.45 + 10.7i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-13.1 + 31.6i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (35.4 + 85.4i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (41.3 - 17.1i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-37.6 + 37.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (52.2 - 52.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 26.9T + 6.24e3T^{2} \)
83 \( 1 + (-10.6 - 25.6i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (103. + 103. i)T + 7.92e3iT^{2} \)
97 \( 1 - 77.9T + 9.40e3T^{2} \)
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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

−16.17666860968696463641567316477, −15.62745563306526914096289554297, −14.41239559253462970938049691538, −13.35313427310362507429797200012, −12.45677699765347080535746958203, −9.723170581960950214267581351321, −8.467396903621889249104208060165, −7.81608413819510064163439460728, −6.02408515740820598320853770376, −3.42285750510656632986763037939, 3.05428787510512237165024686232, 4.15721541689612805112289620141, 7.61488754787857234766419170193, 9.015333638372154875427759326388, 10.00633936203852823256844760757, 11.21271303711703240018100762927, 13.18874574810191204981791724649, 13.71067204990037591086588202251, 15.14629624036278650982976926619, 16.20080446426020293281452430641

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