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Properties

Label 2-2e5-32.3-c2-0-6
Degree $2$
Conductor $32$
Sign $0.406 + 0.913i$
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  + (0.682 − 1.87i)2-s + (0.299 − 0.723i)3-s + (−3.06 − 2.56i)4-s + (1.34 + 3.25i)5-s + (−1.15 − 1.05i)6-s + (0.583 + 0.583i)7-s + (−6.91 + 4.01i)8-s + (5.93 + 5.93i)9-s + (7.03 − 0.313i)10-s + (−3.03 − 7.33i)11-s + (−2.77 + 1.45i)12-s + (−6.38 + 15.4i)13-s + (1.49 − 0.698i)14-s + 2.75·15-s + (2.83 + 15.7i)16-s − 19.0i·17-s + ⋯
L(s)  = 1  + (0.341 − 0.939i)2-s + (0.0999 − 0.241i)3-s + (−0.767 − 0.641i)4-s + (0.269 + 0.650i)5-s + (−0.192 − 0.176i)6-s + (0.0833 + 0.0833i)7-s + (−0.864 + 0.502i)8-s + (0.658 + 0.658i)9-s + (0.703 − 0.0313i)10-s + (−0.276 − 0.666i)11-s + (−0.231 + 0.120i)12-s + (−0.491 + 1.18i)13-s + (0.106 − 0.0498i)14-s + 0.183·15-s + (0.177 + 0.984i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.406 + 0.913i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ 0.406 + 0.913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.956525 - 0.621147i\)
\(L(\frac12)\) \(\approx\) \(0.956525 - 0.621147i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.682 + 1.87i)T \)
good3 \( 1 + (-0.299 + 0.723i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-1.34 - 3.25i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-0.583 - 0.583i)T + 49iT^{2} \)
11 \( 1 + (3.03 + 7.33i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (6.38 - 15.4i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (29.6 + 12.2i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-15.2 + 15.2i)T - 529iT^{2} \)
29 \( 1 + (20.5 + 8.49i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 53.6iT - 961T^{2} \)
37 \( 1 + (3.80 + 9.17i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-14.5 - 14.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (-20.3 - 49.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 4.73T + 2.20e3T^{2} \)
53 \( 1 + (-61.4 + 25.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-42.4 + 17.5i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (27.7 + 11.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (9.42 - 22.7i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (95.1 + 95.1i)T + 5.04e3iT^{2} \)
73 \( 1 + (-37.1 - 37.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 70.3T + 6.24e3T^{2} \)
83 \( 1 + (-14.5 - 6.01i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (60.8 - 60.8i)T - 7.92e3iT^{2} \)
97 \( 1 - 31.8T + 9.40e3T^{2} \)
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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

−16.40008840398295990994509710325, −14.73248513774656417486648461150, −13.81823170767558861749403783121, −12.76632910556365831653460702734, −11.30919147778067388758123079562, −10.35081108772233716003624434577, −8.863142104803881512546899380829, −6.77916897406618574052193176267, −4.71995302524952736081817780980, −2.46178254551422609074999394883, 4.16581363608800844283912144989, 5.73134007399012903517046268322, 7.46364965287028512124201160370, 8.876006446600366265269145891610, 10.18912761542480276184415028782, 12.59846366053924289285138912918, 13.01202082432157903946783062719, 14.93010888876061177331239441328, 15.29168808024467186977812236636, 16.90396586376628832088127415899

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