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Properties

Label 2-2e5-32.21-c3-0-5
Degree $2$
Conductor $32$
Sign $0.700 - 0.713i$
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  + (1.57 + 2.34i)2-s + (4.56 − 1.89i)3-s + (−3.03 + 7.40i)4-s + (1.37 − 3.30i)5-s + (11.6 + 7.75i)6-s + (−6.14 − 6.14i)7-s + (−22.1 + 4.53i)8-s + (−1.79 + 1.79i)9-s + (9.92 − 1.99i)10-s + (17.2 + 7.14i)11-s + (0.141 + 39.5i)12-s + (−25.9 − 62.7i)13-s + (4.75 − 24.1i)14-s − 17.7i·15-s + (−45.5 − 44.9i)16-s − 87.5i·17-s + ⋯
L(s)  = 1  + (0.557 + 0.830i)2-s + (0.879 − 0.364i)3-s + (−0.379 + 0.925i)4-s + (0.122 − 0.295i)5-s + (0.792 + 0.527i)6-s + (−0.331 − 0.331i)7-s + (−0.979 + 0.200i)8-s + (−0.0665 + 0.0665i)9-s + (0.314 − 0.0630i)10-s + (0.472 + 0.195i)11-s + (0.00340 + 0.951i)12-s + (−0.554 − 1.33i)13-s + (0.0907 − 0.460i)14-s − 0.304i·15-s + (−0.712 − 0.702i)16-s − 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.700 - 0.713i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.67851 + 0.704078i\)
\(L(\frac12)\) \(\approx\) \(1.67851 + 0.704078i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.57 - 2.34i)T \)
good3 \( 1 + (-4.56 + 1.89i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-1.37 + 3.30i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (6.14 + 6.14i)T + 343iT^{2} \)
11 \( 1 + (-17.2 - 7.14i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (25.9 + 62.7i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 87.5iT - 4.91e3T^{2} \)
19 \( 1 + (-48.8 - 117. i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (55.7 - 55.7i)T - 1.21e4iT^{2} \)
29 \( 1 + (114. - 47.2i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 229.T + 2.97e4T^{2} \)
37 \( 1 + (123. - 298. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-111. + 111. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-76.5 - 31.7i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 367. iT - 1.03e5T^{2} \)
53 \( 1 + (244. + 101. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (183. - 442. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-524. + 217. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-393. + 162. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (354. + 354. i)T + 3.57e5iT^{2} \)
73 \( 1 + (22.2 - 22.2i)T - 3.89e5iT^{2} \)
79 \( 1 - 396. iT - 4.93e5T^{2} \)
83 \( 1 + (410. + 990. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-170. - 170. i)T + 7.04e5iT^{2} \)
97 \( 1 + 1.72e3T + 9.12e5T^{2} \)
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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

−16.35940399512582601843208437486, −15.07510977560779098145470587008, −14.05400334314506772353863856191, −13.23251039259223547238868289506, −12.04088270477691574246261551606, −9.701662667569945633842622981329, −8.254250683357961899269976720637, −7.23631307301009079013090599319, −5.35076230051458549963351620585, −3.23679190927430065235191939746, 2.59187152824306589752783141034, 4.19277266751318303254441380195, 6.35483288848250376436109082974, 8.852186982178451639673794926599, 9.716356545531333822530081481054, 11.25350812629432453417280951813, 12.49147885204666125428776895252, 13.96782277428918024230928090394, 14.57046594371485811426516539148, 15.72851291487831699138871422969

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