Properties

Label 2-2e5-32.29-c3-0-2
Degree $2$
Conductor $32$
Sign $0.0637 - 0.997i$
Analytic cond. $1.88806$
Root an. cond. $1.37406$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 2.48i)2-s + (−1.36 − 0.564i)3-s + (−4.35 + 6.71i)4-s + (6.58 + 15.8i)5-s + (−0.437 − 4.14i)6-s + (14.5 − 14.5i)7-s + (−22.5 − 1.75i)8-s + (−17.5 − 17.5i)9-s + (−30.6 + 37.8i)10-s + (34.4 − 14.2i)11-s + (9.71 − 6.68i)12-s + (15.3 − 37.1i)13-s + (55.8 + 16.5i)14-s − 25.3i·15-s + (−26.1 − 58.4i)16-s + 103. i·17-s + ⋯
L(s)  = 1  + (0.477 + 0.878i)2-s + (−0.262 − 0.108i)3-s + (−0.544 + 0.839i)4-s + (0.588 + 1.42i)5-s + (−0.0297 − 0.282i)6-s + (0.785 − 0.785i)7-s + (−0.996 − 0.0774i)8-s + (−0.650 − 0.650i)9-s + (−0.967 + 1.19i)10-s + (0.944 − 0.391i)11-s + (0.233 − 0.160i)12-s + (0.328 − 0.792i)13-s + (1.06 + 0.315i)14-s − 0.436i·15-s + (−0.407 − 0.913i)16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0637 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0637 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.0637 - 0.997i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.0637 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07648 + 1.00990i\)
\(L(\frac12)\) \(\approx\) \(1.07648 + 1.00990i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 2.48i)T \)
good3 \( 1 + (1.36 + 0.564i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-6.58 - 15.8i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-14.5 + 14.5i)T - 343iT^{2} \)
11 \( 1 + (-34.4 + 14.2i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-15.3 + 37.1i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 103. iT - 4.91e3T^{2} \)
19 \( 1 + (-12.4 + 29.9i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (72.3 + 72.3i)T + 1.21e4iT^{2} \)
29 \( 1 + (-23.9 - 9.90i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 124.T + 2.97e4T^{2} \)
37 \( 1 + (-18.0 - 43.5i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (45.1 + 45.1i)T + 6.89e4iT^{2} \)
43 \( 1 + (457. - 189. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 582. iT - 1.03e5T^{2} \)
53 \( 1 + (-395. + 163. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-142. - 344. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (34.8 + 14.4i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-196. - 81.4i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (520. - 520. i)T - 3.57e5iT^{2} \)
73 \( 1 + (582. + 582. i)T + 3.89e5iT^{2} \)
79 \( 1 - 157. iT - 4.93e5T^{2} \)
83 \( 1 + (-54.5 + 131. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (272. - 272. i)T - 7.04e5iT^{2} \)
97 \( 1 - 788.T + 9.12e5T^{2} \)
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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

−16.82301497161410554794647614919, −14.89074244785828531420198530729, −14.53228387213281038696868516080, −13.40209183255296137518150696108, −11.67107961872804143947974519509, −10.42965674156481878201121677277, −8.445538861796424767595016099306, −6.85555836878658915767507390057, −5.91388328901737549009875801829, −3.60275456128883665634612702602, 1.76157186159906275393528878914, 4.68806155555872572293587300014, 5.64256991700689255194491857406, 8.691854753765970548044814425950, 9.562031620801547202505906738008, 11.49448823867198561350359238266, 12.06502588971542987610547209721, 13.54053295832821078351600171896, 14.40380526143261091645446538092, 16.13329316284212118538167516096

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