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Properties

Label 2-2e5-32.19-c2-0-4
Degree $2$
Conductor $32$
Sign $0.904 + 0.426i$
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  + (1.82 − 0.820i)2-s + (−0.374 − 0.155i)3-s + (2.65 − 2.99i)4-s + (−7.60 + 3.15i)5-s + (−0.811 + 0.0241i)6-s + (6.84 + 6.84i)7-s + (2.39 − 7.63i)8-s + (−6.24 − 6.24i)9-s + (−11.2 + 11.9i)10-s + (−2.23 + 0.927i)11-s + (−1.46 + 0.709i)12-s + (1.40 + 0.583i)13-s + (18.0 + 6.86i)14-s + 3.34·15-s + (−1.90 − 15.8i)16-s + 2.67i·17-s + ⋯
L(s)  = 1  + (0.912 − 0.410i)2-s + (−0.124 − 0.0517i)3-s + (0.663 − 0.747i)4-s + (−1.52 + 0.630i)5-s + (−0.135 + 0.00402i)6-s + (0.977 + 0.977i)7-s + (0.298 − 0.954i)8-s + (−0.694 − 0.694i)9-s + (−1.12 + 1.19i)10-s + (−0.203 + 0.0842i)11-s + (−0.121 + 0.0591i)12-s + (0.108 + 0.0449i)13-s + (1.29 + 0.490i)14-s + 0.222·15-s + (−0.118 − 0.992i)16-s + 0.157i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.904 + 0.426i$
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: \((2,\ 32,\ (\ :1),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29775 - 0.290332i\)
\(L(\frac12)\) \(\approx\) \(1.29775 - 0.290332i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.82 + 0.820i)T \)
good3 \( 1 + (0.374 + 0.155i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (7.60 - 3.15i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-6.84 - 6.84i)T + 49iT^{2} \)
11 \( 1 + (2.23 - 0.927i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.40 - 0.583i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 2.67iT - 289T^{2} \)
19 \( 1 + (-5.38 + 13.0i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-18.8 + 18.8i)T - 529iT^{2} \)
29 \( 1 + (10.0 - 24.2i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 47.5iT - 961T^{2} \)
37 \( 1 + (28.2 - 11.7i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-6.93 - 6.93i)T + 1.68e3iT^{2} \)
43 \( 1 + (-8.48 + 3.51i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 67.0T + 2.20e3T^{2} \)
53 \( 1 + (10.5 + 25.3i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-27.9 - 67.4i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-31.5 + 76.2i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-90.1 - 37.3i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-1.98 - 1.98i)T + 5.04e3iT^{2} \)
73 \( 1 + (55.5 + 55.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 10.9T + 6.24e3T^{2} \)
83 \( 1 + (-34.1 + 82.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (16.1 - 16.1i)T - 7.92e3iT^{2} \)
97 \( 1 + 62.6T + 9.40e3T^{2} \)
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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

−15.97624096873543737382770955214, −15.00773193485705525265960067312, −14.48720441374664396857983013932, −12.48182422396129836640789984178, −11.62528509490909826198470621404, −10.93784320107748705480382722311, −8.571983839332988318348024922484, −6.87209929754628463303825851602, −5.01369407875409294817245133075, −3.17621368502709902051459804902, 3.94138188045281269372933294009, 5.17096626389097321393372973128, 7.55796276594583304214552525377, 8.162786719097835410949965745040, 11.10163880389321867750127998990, 11.66811875713370374678253173875, 13.16435875337652486536638739612, 14.32475821174384511219469963090, 15.47817754889197860511584546939, 16.51703672695641461856067869178

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