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Properties

Label 2-2e5-32.19-c2-0-1
Degree $2$
Conductor $32$
Sign $0.999 - 0.0133i$
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  + (−1.98 − 0.229i)2-s + (1.58 + 0.655i)3-s + (3.89 + 0.912i)4-s + (4.18 − 1.73i)5-s + (−2.99 − 1.66i)6-s + (3.93 + 3.93i)7-s + (−7.52 − 2.70i)8-s + (−4.29 − 4.29i)9-s + (−8.72 + 2.48i)10-s + (−14.2 + 5.89i)11-s + (5.56 + 3.99i)12-s + (0.454 + 0.188i)13-s + (−6.90 − 8.71i)14-s + 7.76·15-s + (14.3 + 7.11i)16-s − 26.5i·17-s + ⋯
L(s)  = 1  + (−0.993 − 0.114i)2-s + (0.527 + 0.218i)3-s + (0.973 + 0.228i)4-s + (0.837 − 0.346i)5-s + (−0.498 − 0.277i)6-s + (0.561 + 0.561i)7-s + (−0.940 − 0.338i)8-s + (−0.476 − 0.476i)9-s + (−0.872 + 0.248i)10-s + (−1.29 + 0.536i)11-s + (0.463 + 0.332i)12-s + (0.0349 + 0.0144i)13-s + (−0.493 − 0.622i)14-s + 0.517·15-s + (0.895 + 0.444i)16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0133i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.999 - 0.0133i$
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: \((2,\ 32,\ (\ :1),\ 0.999 - 0.0133i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.837616 + 0.00560608i\)
\(L(\frac12)\) \(\approx\) \(0.837616 + 0.00560608i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 + 0.229i)T \)
good3 \( 1 + (-1.58 - 0.655i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-4.18 + 1.73i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-3.93 - 3.93i)T + 49iT^{2} \)
11 \( 1 + (14.2 - 5.89i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-0.454 - 0.188i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 26.5iT - 289T^{2} \)
19 \( 1 + (7.25 - 17.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-0.775 + 0.775i)T - 529iT^{2} \)
29 \( 1 + (17.9 - 43.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 39.6iT - 961T^{2} \)
37 \( 1 + (-36.4 + 15.1i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-38.9 - 38.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (14.2 - 5.91i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 62.1T + 2.20e3T^{2} \)
53 \( 1 + (-11.4 - 27.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (5.30 + 12.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-14.1 + 34.1i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-26.1 - 10.8i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-17.7 - 17.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-12.8 - 12.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 144.T + 6.24e3T^{2} \)
83 \( 1 + (10.9 - 26.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (5.92 - 5.92i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.9T + 9.40e3T^{2} \)
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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

−16.77880127366589639180859833096, −15.53366957523676268571304568281, −14.45537158570948967806430189619, −12.81493501632588738771198913097, −11.40605567638253945705521291758, −9.875711783472621852920811059244, −9.003756046678517026597678926731, −7.71550167765668193579577072450, −5.63507715585375947772100159059, −2.45655708962081033472813390148, 2.36307369042197764591423464299, 5.86781488006479445213776808240, 7.64660541129595065390883347559, 8.627586909193678167648252853369, 10.30763310538865506791195006169, 11.04258061189492680427087019385, 13.17843114085622933776764485291, 14.26240841989088566947639434948, 15.48599711777109890293006031338, 16.98306833445580685305114470415

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