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Properties

Label 2-2e5-32.19-c2-0-2
Degree $2$
Conductor $32$
Sign $0.832 - 0.553i$
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  + (1.61 + 1.17i)2-s + (−1.37 − 0.568i)3-s + (1.23 + 3.80i)4-s + (2.28 − 0.948i)5-s + (−1.55 − 2.53i)6-s + (−6.37 − 6.37i)7-s + (−2.48 + 7.60i)8-s + (−4.80 − 4.80i)9-s + (4.81 + 1.15i)10-s + (−1.79 + 0.744i)11-s + (0.470 − 5.92i)12-s + (16.7 + 6.91i)13-s + (−2.81 − 17.8i)14-s − 3.68·15-s + (−12.9 + 9.38i)16-s + 6.19i·17-s + ⋯
L(s)  = 1  + (0.808 + 0.588i)2-s + (−0.457 − 0.189i)3-s + (0.308 + 0.951i)4-s + (0.457 − 0.189i)5-s + (−0.258 − 0.422i)6-s + (−0.911 − 0.911i)7-s + (−0.310 + 0.950i)8-s + (−0.533 − 0.533i)9-s + (0.481 + 0.115i)10-s + (−0.163 + 0.0676i)11-s + (0.0392 − 0.493i)12-s + (1.28 + 0.532i)13-s + (−0.201 − 1.27i)14-s − 0.245·15-s + (−0.809 + 0.586i)16-s + 0.364i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.832 - 0.553i$
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: \((2,\ 32,\ (\ :1),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21777 + 0.368125i\)
\(L(\frac12)\) \(\approx\) \(1.21777 + 0.368125i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 1.17i)T \)
good3 \( 1 + (1.37 + 0.568i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-2.28 + 0.948i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (6.37 + 6.37i)T + 49iT^{2} \)
11 \( 1 + (1.79 - 0.744i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-16.7 - 6.91i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 6.19iT - 289T^{2} \)
19 \( 1 + (8.50 - 20.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-23.6 + 23.6i)T - 529iT^{2} \)
29 \( 1 + (-14.5 + 35.1i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 14.1iT - 961T^{2} \)
37 \( 1 + (30.0 - 12.4i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (56.9 + 56.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-54.5 + 22.5i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 34.8T + 2.20e3T^{2} \)
53 \( 1 + (-3.92 - 9.48i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-9.41 - 22.7i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-3.00 + 7.25i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (55.9 + 23.1i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-6.27 - 6.27i)T + 5.04e3iT^{2} \)
73 \( 1 + (-66.4 - 66.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.8T + 6.24e3T^{2} \)
83 \( 1 + (1.23 - 2.97i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-36.7 + 36.7i)T - 7.92e3iT^{2} \)
97 \( 1 - 90.0T + 9.40e3T^{2} \)
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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

−16.78325924590349212319823186361, −15.59497484353453788112568535433, −14.10316016300127508076610124566, −13.22998342143913485895313769081, −12.13795991946113939314458447143, −10.62743714638628231806043855553, −8.716881234697943524640094446516, −6.80761872901581286130965559417, −5.87602262442286060735559217227, −3.77313705629048767772651879107, 2.93097737589313784006627683301, 5.31193749824557648425754801388, 6.34476926876568358856640363874, 9.070417423637366354867536539266, 10.52527656843007582664237369831, 11.50975855899925303041048135469, 12.91568171377203232611098118919, 13.75970428112657496256543000648, 15.34166967796570383553233747843, 16.12729685122820323124123404909

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