Properties

Label 2-2e5-32.27-c2-0-1
Degree $2$
Conductor $32$
Sign $-0.276 - 0.960i$
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  + (0.345 + 1.96i)2-s + (−3.70 + 1.53i)3-s + (−3.76 + 1.36i)4-s + (7.20 + 2.98i)5-s + (−4.30 − 6.76i)6-s + (4.26 − 4.26i)7-s + (−3.98 − 6.93i)8-s + (4.99 − 4.99i)9-s + (−3.38 + 15.2i)10-s + (6.19 + 2.56i)11-s + (11.8 − 10.8i)12-s + (−8.05 + 3.33i)13-s + (9.86 + 6.92i)14-s − 31.2·15-s + (12.2 − 10.2i)16-s − 24.5i·17-s + ⋯
L(s)  = 1  + (0.172 + 0.984i)2-s + (−1.23 + 0.511i)3-s + (−0.940 + 0.340i)4-s + (1.44 + 0.596i)5-s + (−0.717 − 1.12i)6-s + (0.608 − 0.608i)7-s + (−0.498 − 0.867i)8-s + (0.554 − 0.554i)9-s + (−0.338 + 1.52i)10-s + (0.563 + 0.233i)11-s + (0.986 − 0.901i)12-s + (−0.619 + 0.256i)13-s + (0.704 + 0.494i)14-s − 2.08·15-s + (0.767 − 0.640i)16-s − 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.276 - 0.960i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ -0.276 - 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.526927 + 0.700025i\)
\(L(\frac12)\) \(\approx\) \(0.526927 + 0.700025i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.345 - 1.96i)T \)
good3 \( 1 + (3.70 - 1.53i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (-7.20 - 2.98i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (-4.26 + 4.26i)T - 49iT^{2} \)
11 \( 1 + (-6.19 - 2.56i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (8.05 - 3.33i)T + (119. - 119. i)T^{2} \)
17 \( 1 + 24.5iT - 289T^{2} \)
19 \( 1 + (-4.96 - 11.9i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (9.72 + 9.72i)T + 529iT^{2} \)
29 \( 1 + (5.86 + 14.1i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 17.5iT - 961T^{2} \)
37 \( 1 + (36.0 + 14.9i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (-10.9 + 10.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (22.4 + 9.27i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 27.0T + 2.20e3T^{2} \)
53 \( 1 + (34.0 - 82.1i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-27.8 + 67.2i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (6.37 + 15.3i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (99.2 - 41.0i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (2.55 - 2.55i)T - 5.04e3iT^{2} \)
73 \( 1 + (-30.7 + 30.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 90.6T + 6.24e3T^{2} \)
83 \( 1 + (39.3 + 94.9i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-109. - 109. i)T + 7.92e3iT^{2} \)
97 \( 1 - 63.7T + 9.40e3T^{2} \)
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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

−17.13402400765891520619589610547, −16.05611276829869164293862877014, −14.41343175323899204022611775443, −13.84190468387846634680475352904, −12.02373909339233748447007593437, −10.46629361047984347400699473475, −9.479265606162735924416786233403, −7.14225651828437331067675097483, −5.92657475439901896895300292065, −4.76576323965420716914321411093, 1.65279032917751645684790179469, 5.14006154752603531368323704510, 6.02451355110132462272179003019, 8.816776980041727076104192510903, 10.15604308906769216211544435344, 11.46388448650432976616207198517, 12.43493162947708457437579729151, 13.34232711157715881256528969680, 14.68831532687983556525255351192, 16.99638525569049725474497495469

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