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Properties

Label 2-88-11.5-c3-0-4
Degree $2$
Conductor $88$
Sign $0.389 + 0.921i$
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  + (−2.34 − 1.70i)3-s + (−2.01 + 6.21i)5-s + (20.0 − 14.5i)7-s + (−5.74 − 17.6i)9-s + (32.9 − 15.7i)11-s + (−22.3 − 68.7i)13-s + (15.3 − 11.1i)15-s + (−1.71 + 5.28i)17-s + (−31.7 − 23.0i)19-s − 71.9·21-s + 172.·23-s + (66.5 + 48.3i)25-s + (−40.8 + 125. i)27-s + (−11.0 + 7.99i)29-s + (−32.5 − 100. i)31-s + ⋯
L(s)  = 1  + (−0.451 − 0.327i)3-s + (−0.180 + 0.555i)5-s + (1.08 − 0.787i)7-s + (−0.212 − 0.655i)9-s + (0.902 − 0.431i)11-s + (−0.476 − 1.46i)13-s + (0.263 − 0.191i)15-s + (−0.0244 + 0.0753i)17-s + (−0.383 − 0.278i)19-s − 0.747·21-s + 1.55·23-s + (0.532 + 0.387i)25-s + (−0.291 + 0.895i)27-s + (−0.0704 + 0.0511i)29-s + (−0.188 − 0.579i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.389 + 0.921i$
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: \((2,\ 88,\ (\ :3/2),\ 0.389 + 0.921i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.10844 - 0.734855i\)
\(L(\frac12)\) \(\approx\) \(1.10844 - 0.734855i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-32.9 + 15.7i)T \)
good3 \( 1 + (2.34 + 1.70i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (2.01 - 6.21i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (-20.0 + 14.5i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (22.3 + 68.7i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (1.71 - 5.28i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (31.7 + 23.0i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 - 172.T + 1.21e4T^{2} \)
29 \( 1 + (11.0 - 7.99i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (32.5 + 100. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (352. - 255. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (53.1 + 38.6i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 230.T + 7.95e4T^{2} \)
47 \( 1 + (198. + 144. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-92.7 - 285. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-71.1 + 51.7i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (103. - 317. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 345.T + 3.00e5T^{2} \)
71 \( 1 + (-203. + 624. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-860. + 624. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-165. - 509. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (9.74 - 29.9i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 1.20e3T + 7.04e5T^{2} \)
97 \( 1 + (161. + 497. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−13.48327619396251668733296095514, −12.27629993087970180732405638543, −11.22478418756824250995112076541, −10.57534767381386964986350783025, −8.919492571572108113081388676510, −7.58485766236519429234647669243, −6.59903691374715700684785638120, −5.10116811530944817512523194945, −3.38149642647001139748849088613, −0.932410038234645167240944775196, 1.88416715320367914411331595462, 4.46207636099811113800106520716, 5.24295221652677244978595091511, 6.93229181848011103758389616605, 8.472763766129043200161368335204, 9.274725401750217466290569464635, 10.88194119105149220789590992283, 11.69864774083889686454124050031, 12.50021731241442672317920657727, 14.11890824432242874803531652889

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