Properties

Label 2-2e5-32.21-c7-0-21
Degree $2$
Conductor $32$
Sign $-0.410 + 0.911i$
Analytic cond. $9.99632$
Root an. cond. $3.16169$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.374 + 11.3i)2-s + (19.9 − 8.26i)3-s + (−127. + 8.47i)4-s + (−34.6 + 83.5i)5-s + (100. + 222. i)6-s + (−609. − 609. i)7-s + (−143. − 1.44e3i)8-s + (−1.21e3 + 1.21e3i)9-s + (−958. − 360. i)10-s + (−3.76e3 − 1.56e3i)11-s + (−2.47e3 + 1.22e3i)12-s + (−450. − 1.08e3i)13-s + (6.66e3 − 7.12e3i)14-s + 1.95e3i·15-s + (1.62e4 − 2.16e3i)16-s − 1.74e4i·17-s + ⋯
L(s)  = 1  + (0.0331 + 0.999i)2-s + (0.426 − 0.176i)3-s + (−0.997 + 0.0662i)4-s + (−0.123 + 0.299i)5-s + (0.190 + 0.420i)6-s + (−0.671 − 0.671i)7-s + (−0.0992 − 0.995i)8-s + (−0.556 + 0.556i)9-s + (−0.302 − 0.113i)10-s + (−0.853 − 0.353i)11-s + (−0.414 + 0.204i)12-s + (−0.0569 − 0.137i)13-s + (0.649 − 0.693i)14-s + 0.149i·15-s + (0.991 − 0.132i)16-s − 0.860i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.410 + 0.911i$
Analytic conductor: \(9.99632\)
Root analytic conductor: \(3.16169\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :7/2),\ -0.410 + 0.911i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0484131 - 0.0749174i\)
\(L(\frac12)\) \(\approx\) \(0.0484131 - 0.0749174i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.374 - 11.3i)T \)
good3 \( 1 + (-19.9 + 8.26i)T + (1.54e3 - 1.54e3i)T^{2} \)
5 \( 1 + (34.6 - 83.5i)T + (-5.52e4 - 5.52e4i)T^{2} \)
7 \( 1 + (609. + 609. i)T + 8.23e5iT^{2} \)
11 \( 1 + (3.76e3 + 1.56e3i)T + (1.37e7 + 1.37e7i)T^{2} \)
13 \( 1 + (450. + 1.08e3i)T + (-4.43e7 + 4.43e7i)T^{2} \)
17 \( 1 + 1.74e4iT - 4.10e8T^{2} \)
19 \( 1 + (1.03e4 + 2.50e4i)T + (-6.32e8 + 6.32e8i)T^{2} \)
23 \( 1 + (2.30e4 - 2.30e4i)T - 3.40e9iT^{2} \)
29 \( 1 + (1.63e5 - 6.78e4i)T + (1.21e10 - 1.21e10i)T^{2} \)
31 \( 1 + 7.35e4T + 2.75e10T^{2} \)
37 \( 1 + (6.54e4 - 1.58e5i)T + (-6.71e10 - 6.71e10i)T^{2} \)
41 \( 1 + (4.54e5 - 4.54e5i)T - 1.94e11iT^{2} \)
43 \( 1 + (-4.74e5 - 1.96e5i)T + (1.92e11 + 1.92e11i)T^{2} \)
47 \( 1 + 4.05e3iT - 5.06e11T^{2} \)
53 \( 1 + (3.14e5 + 1.30e5i)T + (8.30e11 + 8.30e11i)T^{2} \)
59 \( 1 + (-8.03e5 + 1.93e6i)T + (-1.75e12 - 1.75e12i)T^{2} \)
61 \( 1 + (1.88e6 - 7.79e5i)T + (2.22e12 - 2.22e12i)T^{2} \)
67 \( 1 + (-3.02e6 + 1.25e6i)T + (4.28e12 - 4.28e12i)T^{2} \)
71 \( 1 + (-6.25e5 - 6.25e5i)T + 9.09e12iT^{2} \)
73 \( 1 + (-1.17e6 + 1.17e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 6.82e6iT - 1.92e13T^{2} \)
83 \( 1 + (-2.54e6 - 6.13e6i)T + (-1.91e13 + 1.91e13i)T^{2} \)
89 \( 1 + (-6.78e5 - 6.78e5i)T + 4.42e13iT^{2} \)
97 \( 1 + 8.54e6T + 8.07e13T^{2} \)
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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

−14.90491433000816066292580343505, −13.70689087385877457673775256902, −13.06000278284622933021626820389, −10.90649505818037209550780484355, −9.388750872738840867123831707191, −7.975264838424194158399994354267, −6.92699374353096033327089556876, −5.22134010301038943801851345025, −3.21402121935425377345870425129, −0.03745625404031204621533010066, 2.34174041540394738079156086363, 3.83417799108647315822766168839, 5.73184829281500559227605991640, 8.297460457153348293969350642248, 9.321576213266817449173477281043, 10.53542415045562046591527462524, 12.14636623387709145307357149810, 12.84598513467839426826774683680, 14.31938141433951300471275662955, 15.41304528105806832221376776748

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