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Properties

Label 2-88-11.5-c3-0-6
Degree $2$
Conductor $88$
Sign $0.696 + 0.717i$
Analytic cond. $5.19216$
Root an. cond. $2.27863$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.61 + 1.89i)3-s + (6.32 − 19.4i)5-s + (−5.38 + 3.91i)7-s + (−5.12 − 15.7i)9-s + (25.0 − 26.5i)11-s + (16.3 + 50.2i)13-s + (53.4 − 38.8i)15-s + (18.3 − 56.5i)17-s + (89.7 + 65.1i)19-s − 21.4·21-s − 48.0·23-s + (−237. − 172. i)25-s + (43.4 − 133. i)27-s + (−100. + 72.8i)29-s + (94.2 + 290. i)31-s + ⋯
L(s)  = 1  + (0.502 + 0.365i)3-s + (0.565 − 1.74i)5-s + (−0.290 + 0.211i)7-s + (−0.189 − 0.583i)9-s + (0.685 − 0.728i)11-s + (0.348 + 1.07i)13-s + (0.920 − 0.668i)15-s + (0.262 − 0.807i)17-s + (1.08 + 0.787i)19-s − 0.223·21-s − 0.435·23-s + (−1.90 − 1.38i)25-s + (0.309 − 0.953i)27-s + (−0.642 + 0.466i)29-s + (0.546 + 1.68i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(5.19216\)
Root analytic conductor: \(2.27863\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :3/2),\ 0.696 + 0.717i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.72877 - 0.731669i\)
\(L(\frac12)\) \(\approx\) \(1.72877 - 0.731669i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-25.0 + 26.5i)T \)
good3 \( 1 + (-2.61 - 1.89i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (-6.32 + 19.4i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (5.38 - 3.91i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (-16.3 - 50.2i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-18.3 + 56.5i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-89.7 - 65.1i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 48.0T + 1.21e4T^{2} \)
29 \( 1 + (100. - 72.8i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-94.2 - 290. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (128. - 93.2i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (-184. - 134. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 302.T + 7.95e4T^{2} \)
47 \( 1 + (0.196 + 0.142i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (41.2 + 127. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (227. - 165. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-165. + 509. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 - 695.T + 3.00e5T^{2} \)
71 \( 1 + (118. - 365. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-616. + 447. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (71.6 + 220. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (232. - 715. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + (-20.9 - 64.4i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−13.75188349945266998225829570537, −12.39622584825392848840094784187, −11.71285684289257954415221849538, −9.688479053128394346604318479017, −9.167555158951180392862646882100, −8.354370289888708530354989727082, −6.32367285345193611624001297397, −5.05446431333636937400022819176, −3.58618931059507346504485670572, −1.25109659199712826329364511889, 2.22126717945734530868991645545, 3.47151758470875997607260210357, 5.81116210793122571381739190435, 7.00147310330274712492604684628, 7.88213885627699402606765571388, 9.622548937219314714780555045846, 10.47381071635186547433024954126, 11.48043727510914162221714789082, 13.09580736465596685697589533463, 13.84719891030584087409180773818

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