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Properties

Label 2-88-88.27-c4-0-26
Degree $2$
Conductor $88$
Sign $0.648 + 0.761i$
Analytic cond. $9.09655$
Root an. cond. $3.01604$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 + 3.80i)2-s + (−5.20 − 3.78i)3-s + (−12.9 + 9.40i)4-s + (7.95 − 24.4i)6-s + (−51.7 − 37.6i)8-s + (−12.2 − 37.6i)9-s + (88.4 − 82.5i)11-s + 102.·12-s + (79.1 − 243. i)16-s + (155. − 479. i)17-s + (128. − 93.1i)18-s + (9.67 + 7.03i)19-s + (423. + 234. i)22-s + (127. + 391. i)24-s + (−505. − 367. i)25-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.578 − 0.420i)3-s + (−0.809 + 0.587i)4-s + (0.220 − 0.679i)6-s + (−0.809 − 0.587i)8-s + (−0.151 − 0.465i)9-s + (0.730 − 0.682i)11-s + 0.714·12-s + (0.309 − 0.951i)16-s + (0.539 − 1.65i)17-s + (0.395 − 0.287i)18-s + (0.0268 + 0.0194i)19-s + (0.874 + 0.484i)22-s + (0.220 + 0.679i)24-s + (−0.809 − 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.648 + 0.761i$
Analytic conductor: \(9.09655\)
Root analytic conductor: \(3.01604\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :2),\ 0.648 + 0.761i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.935288 - 0.431960i\)
\(L(\frac12)\) \(\approx\) \(0.935288 - 0.431960i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 3.80i)T \)
11 \( 1 + (-88.4 + 82.5i)T \)
good3 \( 1 + (5.20 + 3.78i)T + (25.0 + 77.0i)T^{2} \)
5 \( 1 + (505. + 367. i)T^{2} \)
7 \( 1 + (-741. + 2.28e3i)T^{2} \)
13 \( 1 + (2.31e4 - 1.67e4i)T^{2} \)
17 \( 1 + (-155. + 479. i)T + (-6.75e4 - 4.90e4i)T^{2} \)
19 \( 1 + (-9.67 - 7.03i)T + (4.02e4 + 1.23e5i)T^{2} \)
23 \( 1 - 2.79e5T^{2} \)
29 \( 1 + (-2.18e5 + 6.72e5i)T^{2} \)
31 \( 1 + (7.47e5 - 5.42e5i)T^{2} \)
37 \( 1 + (-5.79e5 + 1.78e6i)T^{2} \)
41 \( 1 + (2.09e3 + 1.51e3i)T + (8.73e5 + 2.68e6i)T^{2} \)
43 \( 1 + 2.21e3T + 3.41e6T^{2} \)
47 \( 1 + (-1.50e6 - 4.64e6i)T^{2} \)
53 \( 1 + (6.38e6 - 4.63e6i)T^{2} \)
59 \( 1 + (3.46e3 - 2.51e3i)T + (3.74e6 - 1.15e7i)T^{2} \)
61 \( 1 + (1.12e7 + 8.13e6i)T^{2} \)
67 \( 1 - 8.48e3T + 2.01e7T^{2} \)
71 \( 1 + (2.05e7 + 1.49e7i)T^{2} \)
73 \( 1 + (-1.33e3 + 967. i)T + (8.77e6 - 2.70e7i)T^{2} \)
79 \( 1 + (3.15e7 - 2.28e7i)T^{2} \)
83 \( 1 + (1.29e3 - 3.98e3i)T + (-3.83e7 - 2.78e7i)T^{2} \)
89 \( 1 - 1.58e4T + 6.27e7T^{2} \)
97 \( 1 + (5.64e3 + 1.73e4i)T + (-7.16e7 + 5.20e7i)T^{2} \)
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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

−13.53382621507152315083626235537, −12.17795671575693657149953482499, −11.59640973807746220660408186732, −9.705037621576472227526382332228, −8.610571508278536735609574461040, −7.22754898944700856502231720351, −6.29566972923109321044511581271, −5.20533376786201572380194390355, −3.50575267494752434666942627600, −0.50208729080696119987322360123, 1.72817994323419724176108238975, 3.72875331657343385969811507061, 4.94880012874282611074601569698, 6.17077364207392496226094047554, 8.175511643445787503806459047285, 9.613669959958360176694676748145, 10.45287252501240410416641441403, 11.42481984426347225413708970692, 12.31597338709382909582349466066, 13.37554395300911761552621842817

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