Properties

Label 2-2e5-32.3-c2-0-4
Degree $2$
Conductor $32$
Sign $0.925 + 0.379i$
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  + (−1.46 + 1.35i)2-s + (2.10 − 5.07i)3-s + (0.309 − 3.98i)4-s + (1.74 + 4.21i)5-s + (3.80 + 10.3i)6-s + (−0.392 − 0.392i)7-s + (4.96 + 6.27i)8-s + (−14.9 − 14.9i)9-s + (−8.29 − 3.81i)10-s + (2.90 + 7.02i)11-s + (−19.5 − 9.95i)12-s + (−4.50 + 10.8i)13-s + (1.10 + 0.0429i)14-s + 25.0·15-s + (−15.8 − 2.46i)16-s + 10.5i·17-s + ⋯
L(s)  = 1  + (−0.733 + 0.679i)2-s + (0.700 − 1.69i)3-s + (0.0772 − 0.997i)4-s + (0.349 + 0.843i)5-s + (0.634 + 1.71i)6-s + (−0.0560 − 0.0560i)7-s + (0.620 + 0.784i)8-s + (−1.66 − 1.66i)9-s + (−0.829 − 0.381i)10-s + (0.264 + 0.638i)11-s + (−1.63 − 0.829i)12-s + (−0.346 + 0.836i)13-s + (0.0792 + 0.00306i)14-s + 1.67·15-s + (−0.988 − 0.154i)16-s + 0.620i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.925 + 0.379i$
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: \((2,\ 32,\ (\ :1),\ 0.925 + 0.379i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.865643 - 0.170835i\)
\(L(\frac12)\) \(\approx\) \(0.865643 - 0.170835i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.46 - 1.35i)T \)
good3 \( 1 + (-2.10 + 5.07i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-1.74 - 4.21i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (0.392 + 0.392i)T + 49iT^{2} \)
11 \( 1 + (-2.90 - 7.02i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (4.50 - 10.8i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 10.5iT - 289T^{2} \)
19 \( 1 + (1.88 + 0.781i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (0.445 - 0.445i)T - 529iT^{2} \)
29 \( 1 + (-0.741 - 0.307i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 47.6iT - 961T^{2} \)
37 \( 1 + (-14.5 - 35.0i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (11.3 + 11.3i)T + 1.68e3iT^{2} \)
43 \( 1 + (14.6 + 35.3i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 80.5T + 2.20e3T^{2} \)
53 \( 1 + (66.6 - 27.5i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-65.0 + 26.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-87.4 - 36.2i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (7.12 - 17.1i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (14.8 + 14.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (-18.6 - 18.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 36.2T + 6.24e3T^{2} \)
83 \( 1 + (-27.0 - 11.2i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (56.4 - 56.4i)T - 7.92e3iT^{2} \)
97 \( 1 - 158.T + 9.40e3T^{2} \)
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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

−16.98481732903970448616218446634, −14.94720478903031442934515936608, −14.31790805975848871246872124182, −13.20073484167474110420668015775, −11.61002317399512354926906365199, −9.778115405911324973689308132119, −8.341259671726478146067360136763, −7.10514832812135039619032661397, −6.37817601642710346285054049975, −2.05590251460062945785704805259, 3.20936813046540260023704168544, 4.92146256886603644578690108329, 8.265384662239433217877938028687, 9.178258874500397801103760337072, 10.05906556408778074513518198921, 11.23834401345801972996154279877, 12.94467596810521957861839441768, 14.38068684906211839532606332485, 15.91700694169994988462309512106, 16.50713496598866133573351137848

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