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Properties

Label 2-2e5-32.21-c1-0-1
Degree $2$
Conductor $32$
Sign $0.999 - 0.00437i$
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  + (−1.40 + 0.167i)2-s + (1.27 − 0.529i)3-s + (1.94 − 0.470i)4-s + (−0.707 + 1.70i)5-s + (−1.70 + 0.957i)6-s + (−2.74 − 2.74i)7-s + (−2.65 + 0.985i)8-s + (−0.766 + 0.766i)9-s + (0.707 − 2.51i)10-s + (0.135 + 0.0560i)11-s + (2.23 − 1.63i)12-s + (1.18 + 2.85i)13-s + (4.32 + 3.40i)14-s + 2.55i·15-s + (3.55 − 1.82i)16-s − 6.44i·17-s + ⋯
L(s)  = 1  + (−0.992 + 0.118i)2-s + (0.738 − 0.305i)3-s + (0.971 − 0.235i)4-s + (−0.316 + 0.763i)5-s + (−0.696 + 0.391i)6-s + (−1.03 − 1.03i)7-s + (−0.937 + 0.348i)8-s + (−0.255 + 0.255i)9-s + (0.223 − 0.795i)10-s + (0.0408 + 0.0169i)11-s + (0.645 − 0.470i)12-s + (0.327 + 0.790i)13-s + (1.15 + 0.908i)14-s + 0.660i·15-s + (0.889 − 0.457i)16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.999 - 0.00437i$
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: \((2,\ 32,\ (\ :1/2),\ 0.999 - 0.00437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545990 + 0.00119456i\)
\(L(\frac12)\) \(\approx\) \(0.545990 + 0.00119456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.167i)T \)
good3 \( 1 + (-1.27 + 0.529i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.707 - 1.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.74 + 2.74i)T + 7iT^{2} \)
11 \( 1 + (-0.135 - 0.0560i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.18 - 2.85i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 6.44iT - 17T^{2} \)
19 \( 1 + (-0.805 - 1.94i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.749 + 0.749i)T - 23iT^{2} \)
29 \( 1 + (-4.32 + 1.79i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (-1.73 + 4.18i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.49 - 2.49i)T - 41iT^{2} \)
43 \( 1 + (6.10 + 2.52i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 2.66iT - 47T^{2} \)
53 \( 1 + (-1.64 - 0.682i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.43 + 3.47i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.46 - 1.43i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (14.0 - 5.83i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (3.40 + 3.40i)T + 71iT^{2} \)
73 \( 1 + (0.442 - 0.442i)T - 73iT^{2} \)
79 \( 1 + 7.07iT - 79T^{2} \)
83 \( 1 + (-2.99 - 7.23i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.21 + 4.21i)T + 89iT^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−16.78822038716304393768201487142, −15.99356456208478793607533818136, −14.50867896782748005174705761216, −13.53730427185499292584900409897, −11.60786617859072439796066293445, −10.36775388008133967301600583236, −9.140540461896382787966412353491, −7.57790062811542124808616291238, −6.71070073792761251293521555067, −3.06014740097946193535272357069, 3.14701352631460913060268324442, 6.16220400041491996203972656184, 8.340579096666816766094023682827, 8.930149116225319517445708874826, 10.17521481345542911491279484940, 11.96726639454369429514390160547, 12.93191992295355812935202824149, 15.08321500145372671801757480710, 15.68551673670633767509785236264, 16.84670491316641671520896865802

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