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Properties

Label 2-2e5-32.21-c1-0-2
Degree $2$
Conductor $32$
Sign $0.922 + 0.386i$
Analytic cond. $0.255521$
Root an. cond. $0.505491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 − 0.874i)2-s + (−2.27 + 0.943i)3-s + (0.470 − 1.94i)4-s + (−0.707 + 1.70i)5-s + (−1.70 + 3.04i)6-s + (−0.665 − 0.665i)7-s + (−1.17 − 2.57i)8-s + (2.18 − 2.18i)9-s + (0.707 + 2.51i)10-s + (3.69 + 1.52i)11-s + (0.763 + 4.87i)12-s + (−1.76 − 4.26i)13-s + (−1.32 − 0.157i)14-s − 4.55i·15-s + (−3.55 − 1.82i)16-s + 3.61i·17-s + ⋯
L(s)  = 1  + (0.785 − 0.618i)2-s + (−1.31 + 0.544i)3-s + (0.235 − 0.971i)4-s + (−0.316 + 0.763i)5-s + (−0.696 + 1.24i)6-s + (−0.251 − 0.251i)7-s + (−0.416 − 0.909i)8-s + (0.726 − 0.726i)9-s + (0.223 + 0.795i)10-s + (1.11 + 0.461i)11-s + (0.220 + 1.40i)12-s + (−0.489 − 1.18i)13-s + (−0.353 − 0.0420i)14-s − 1.17i·15-s + (−0.889 − 0.457i)16-s + 0.877i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(0.255521\)
Root analytic conductor: \(0.505491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1/2),\ 0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713010 - 0.143448i\)
\(L(\frac12)\) \(\approx\) \(0.713010 - 0.143448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.874i)T \)
good3 \( 1 + (2.27 - 0.943i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.707 - 1.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.665 + 0.665i)T + 7iT^{2} \)
11 \( 1 + (-3.69 - 1.52i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.76 + 4.26i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 + (-0.194 - 0.470i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.33 - 1.33i)T - 23iT^{2} \)
29 \( 1 + (5.73 - 2.37i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (-0.510 + 1.23i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.66 + 1.66i)T - 41iT^{2} \)
43 \( 1 + (2.54 + 1.05i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.49iT - 47T^{2} \)
53 \( 1 + (-4.59 - 1.90i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.04 + 4.94i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-13.7 + 5.67i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.40 - 1.41i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (9.66 + 9.66i)T + 71iT^{2} \)
73 \( 1 + (7.55 - 7.55i)T - 73iT^{2} \)
79 \( 1 - 17.2iT - 79T^{2} \)
83 \( 1 + (-4.82 - 11.6i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.43 + 5.43i)T + 89iT^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−16.78786468354139161935767937362, −15.39679359572117099195523493536, −14.58604792315740484317845175714, −12.81382496268897251505247202068, −11.74078793101790864242530557448, −10.78455042508734454121790246804, −9.915151549809928104699173058312, −6.83252945163110722676922109156, −5.50476891258354297994161463185, −3.82778364066925301051582187499, 4.53522746091494322441245469800, 5.98790928137720585457857927396, 7.10000703682061649031630814747, 8.980040437635186735933173768218, 11.61762659939068749117182255641, 11.94121058390585335074345342706, 13.15781485895449950610490404918, 14.51084980504463418717684152959, 16.26846776735536585740941158125, 16.61365290460613127083666213129

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