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Properties

Label 2-2e5-32.19-c2-0-0
Degree $2$
Conductor $32$
Sign $0.312 - 0.949i$
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  + (−0.360 + 1.96i)2-s + (2.49 + 1.03i)3-s + (−3.73 − 1.41i)4-s + (−0.452 + 0.187i)5-s + (−2.93 + 4.53i)6-s + (0.429 + 0.429i)7-s + (4.14 − 6.84i)8-s + (−1.19 − 1.19i)9-s + (−0.205 − 0.957i)10-s + (17.3 − 7.18i)11-s + (−7.86 − 7.41i)12-s + (−19.9 − 8.26i)13-s + (−1.00 + 0.690i)14-s − 1.32·15-s + (11.9 + 10.6i)16-s + 13.5i·17-s + ⋯
L(s)  = 1  + (−0.180 + 0.983i)2-s + (0.832 + 0.344i)3-s + (−0.934 − 0.354i)4-s + (−0.0904 + 0.0374i)5-s + (−0.489 + 0.756i)6-s + (0.0614 + 0.0614i)7-s + (0.517 − 0.855i)8-s + (−0.133 − 0.133i)9-s + (−0.0205 − 0.0957i)10-s + (1.57 − 0.652i)11-s + (−0.655 − 0.617i)12-s + (−1.53 − 0.635i)13-s + (−0.0714 + 0.0493i)14-s − 0.0882·15-s + (0.747 + 0.663i)16-s + 0.799i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.312 - 0.949i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ 0.312 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.844035 + 0.610648i\)
\(L(\frac12)\) \(\approx\) \(0.844035 + 0.610648i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.360 - 1.96i)T \)
good3 \( 1 + (-2.49 - 1.03i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.452 - 0.187i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-0.429 - 0.429i)T + 49iT^{2} \)
11 \( 1 + (-17.3 + 7.18i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (19.9 + 8.26i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 13.5iT - 289T^{2} \)
19 \( 1 + (3.45 - 8.34i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (16.8 - 16.8i)T - 529iT^{2} \)
29 \( 1 + (-13.8 + 33.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 24.5iT - 961T^{2} \)
37 \( 1 + (-9.89 + 4.09i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-14.4 - 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (-17.8 + 7.39i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 43.6T + 2.20e3T^{2} \)
53 \( 1 + (-28.0 - 67.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-1.70 - 4.10i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-3.53 + 8.53i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (0.300 + 0.124i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (29.0 + 29.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (68.2 + 68.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 67.7T + 6.24e3T^{2} \)
83 \( 1 + (16.4 - 39.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (45.3 - 45.3i)T - 7.92e3iT^{2} \)
97 \( 1 + 119.T + 9.40e3T^{2} \)
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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

−16.89150639141384729422954338725, −15.37898635483016397433316228948, −14.65047172215173706643027273445, −13.81076019672503983454982379521, −12.06823175869098702527451604462, −9.943819741442571779654539249601, −8.934926843954843830041571193246, −7.73477348787961148403041400662, −6.00089047485615853975053698047, −3.90255425921258343796541611581, 2.32548883631757481471812895439, 4.39978942781689190287612079515, 7.32357019631110641216283871430, 8.840451674379023619253627078378, 9.804317309264955915405848851760, 11.59174082864454174303821665671, 12.48719797132526451463916850218, 14.05074454321922415567262874089, 14.52545261970177613749119107524, 16.71963171206596534686266141715

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