Properties

Label 2-2e5-32.19-c2-0-5
Degree $2$
Conductor $32$
Sign $-0.847 + 0.531i$
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  + (−1.62 + 1.16i)2-s + (−4.68 − 1.94i)3-s + (1.27 − 3.78i)4-s + (−4.51 + 1.86i)5-s + (9.87 − 2.31i)6-s + (−3.85 − 3.85i)7-s + (2.34 + 7.65i)8-s + (11.8 + 11.8i)9-s + (5.14 − 8.29i)10-s + (−4.56 + 1.89i)11-s + (−13.3 + 15.2i)12-s + (−5.58 − 2.31i)13-s + (10.7 + 1.76i)14-s + 24.7·15-s + (−12.7 − 9.70i)16-s − 25.0i·17-s + ⋯
L(s)  = 1  + (−0.812 + 0.583i)2-s + (−1.56 − 0.647i)3-s + (0.319 − 0.947i)4-s + (−0.902 + 0.373i)5-s + (1.64 − 0.385i)6-s + (−0.550 − 0.550i)7-s + (0.292 + 0.956i)8-s + (1.31 + 1.31i)9-s + (0.514 − 0.829i)10-s + (−0.414 + 0.171i)11-s + (−1.11 + 1.27i)12-s + (−0.429 − 0.177i)13-s + (0.768 + 0.126i)14-s + 1.65·15-s + (−0.795 − 0.606i)16-s − 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.847 + 0.531i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ -0.847 + 0.531i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0265890 - 0.0924223i\)
\(L(\frac12)\) \(\approx\) \(0.0265890 - 0.0924223i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.62 - 1.16i)T \)
good3 \( 1 + (4.68 + 1.94i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (4.51 - 1.86i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (3.85 + 3.85i)T + 49iT^{2} \)
11 \( 1 + (4.56 - 1.89i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (5.58 + 2.31i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 25.0iT - 289T^{2} \)
19 \( 1 + (-6.43 + 15.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (26.9 - 26.9i)T - 529iT^{2} \)
29 \( 1 + (0.210 - 0.507i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 15.8iT - 961T^{2} \)
37 \( 1 + (2.18 - 0.905i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (31.1 + 31.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (-12.9 + 5.34i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 15.0T + 2.20e3T^{2} \)
53 \( 1 + (15.4 + 37.2i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (14.7 + 35.5i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (15.4 - 37.3i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-61.3 - 25.4i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (51.7 + 51.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-64.9 - 64.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 38.1T + 6.24e3T^{2} \)
83 \( 1 + (-15.9 + 38.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-23.7 + 23.7i)T - 7.92e3iT^{2} \)
97 \( 1 + 118.T + 9.40e3T^{2} \)
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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

−16.22570218090013295723565238917, −15.60149828352961656094268973054, −13.65588614386520164553360823419, −11.96636832530569192086634224194, −11.13207844218318692664463509653, −9.896615939571489765820980759729, −7.53234680993026455970787555760, −6.90558411161663554578036624369, −5.27723498960608772891195883872, −0.17693124480114067031269225390, 4.14288406269716082451246696331, 6.16122977039455051458623055301, 8.142993328345878188769520666111, 9.844604523252091589817594598515, 10.82301800542092900072673255385, 12.05929875179763094506645616308, 12.49810640749717450486286371887, 15.43944036301911306986813855671, 16.27828464318368103230785942523, 16.94088429818498239783085013889

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