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Properties

Label 2-2e5-32.29-c3-0-8
Degree $2$
Conductor $32$
Sign $-0.691 + 0.722i$
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  + (−0.755 + 2.72i)2-s + (−6.06 − 2.51i)3-s + (−6.85 − 4.12i)4-s + (−2.91 − 7.02i)5-s + (11.4 − 14.6i)6-s + (−13.3 + 13.3i)7-s + (16.4 − 15.5i)8-s + (11.3 + 11.3i)9-s + (21.3 − 2.62i)10-s + (−49.6 + 20.5i)11-s + (31.2 + 42.2i)12-s + (8.74 − 21.1i)13-s + (−26.3 − 46.6i)14-s + 49.9i·15-s + (30.0 + 56.5i)16-s + 77.7i·17-s + ⋯
L(s)  = 1  + (−0.267 + 0.963i)2-s + (−1.16 − 0.483i)3-s + (−0.857 − 0.515i)4-s + (−0.260 − 0.628i)5-s + (0.777 − 0.995i)6-s + (−0.722 + 0.722i)7-s + (0.725 − 0.688i)8-s + (0.421 + 0.421i)9-s + (0.675 − 0.0829i)10-s + (−1.36 + 0.563i)11-s + (0.751 + 1.01i)12-s + (0.186 − 0.450i)13-s + (−0.503 − 0.889i)14-s + 0.859i·15-s + (0.469 + 0.882i)16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.691 + 0.722i$
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: \((2,\ 32,\ (\ :3/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0319091 - 0.0747274i\)
\(L(\frac12)\) \(\approx\) \(0.0319091 - 0.0747274i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 - 2.72i)T \)
good3 \( 1 + (6.06 + 2.51i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (2.91 + 7.02i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (13.3 - 13.3i)T - 343iT^{2} \)
11 \( 1 + (49.6 - 20.5i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-8.74 + 21.1i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 77.7iT - 4.91e3T^{2} \)
19 \( 1 + (-53.3 + 128. i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (35.5 + 35.5i)T + 1.21e4iT^{2} \)
29 \( 1 + (245. + 101. i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 202.T + 2.97e4T^{2} \)
37 \( 1 + (-36.3 - 87.6i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (36.8 + 36.8i)T + 6.89e4iT^{2} \)
43 \( 1 + (-185. + 76.8i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 82.9iT - 1.03e5T^{2} \)
53 \( 1 + (534. - 221. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-75.7 - 182. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (472. + 195. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (102. + 42.5i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-520. + 520. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-244. - 244. i)T + 3.89e5iT^{2} \)
79 \( 1 + 774. iT - 4.93e5T^{2} \)
83 \( 1 + (23.9 - 57.7i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (351. - 351. i)T - 7.04e5iT^{2} \)
97 \( 1 - 1.30e3T + 9.12e5T^{2} \)
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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

−15.97868210838071726149445976905, −15.23059017449504949833314302263, −13.08186227658552656553669471795, −12.55501535411270083361217668379, −10.77262880688700435607144120548, −9.176179783829176966347026380563, −7.64051445233594664581891749178, −6.13060148980386429666751811992, −5.10100768813809695846295371440, −0.090281794308423242625109735734, 3.50408779275038726303963470438, 5.42387933691129082956022718136, 7.52322445693188180358381630338, 9.694012070075674731534711966755, 10.72224814282908401517416544171, 11.37380786848717004082448698217, 12.80623145482441859294011021935, 14.07008842159620369026005862650, 16.15523614853218449906628547708, 16.66428629298571469384283370880

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