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Properties

Label 2-88-88.19-c1-0-3
Degree $2$
Conductor $88$
Sign $0.727 + 0.685i$
Analytic cond. $0.702683$
Root an. cond. $0.838262$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.329i)2-s + (−0.303 − 0.935i)3-s + (1.78 − 0.906i)4-s + (−0.398 + 0.548i)5-s + (0.726 + 1.18i)6-s + (1.40 − 4.32i)7-s + (−2.15 + 1.83i)8-s + (1.64 − 1.19i)9-s + (0.367 − 0.885i)10-s + (3.31 − 0.132i)11-s + (−1.38 − 1.39i)12-s + (−1.90 + 1.38i)13-s + (−0.506 + 6.40i)14-s + (0.633 + 0.205i)15-s + (2.35 − 3.23i)16-s + (−2.07 + 2.85i)17-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)2-s + (−0.175 − 0.539i)3-s + (0.891 − 0.453i)4-s + (−0.178 + 0.245i)5-s + (0.296 + 0.484i)6-s + (0.530 − 1.63i)7-s + (−0.761 + 0.648i)8-s + (0.548 − 0.398i)9-s + (0.116 − 0.279i)10-s + (0.999 − 0.0400i)11-s + (−0.401 − 0.401i)12-s + (−0.528 + 0.383i)13-s + (−0.135 + 1.71i)14-s + (0.163 + 0.0531i)15-s + (0.588 − 0.808i)16-s + (−0.503 + 0.692i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.727 + 0.685i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.727 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611581 - 0.242669i\)
\(L(\frac12)\) \(\approx\) \(0.611581 - 0.242669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.329i)T \)
11 \( 1 + (-3.31 + 0.132i)T \)
good3 \( 1 + (0.303 + 0.935i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.398 - 0.548i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.40 + 4.32i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.90 - 1.38i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.07 - 2.85i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.38 - 1.42i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.11iT - 23T^{2} \)
29 \( 1 + (0.379 - 1.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.46 - 4.76i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.78 - 2.85i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.88 + 0.612i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 + (4.48 - 1.45i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.70 + 6.46i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.400 - 1.23i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.73 - 4.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.483T + 67T^{2} \)
71 \( 1 + (8.68 - 11.9i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.79 - 2.85i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.848 - 0.616i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.37 - 6.02i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.47T + 89T^{2} \)
97 \( 1 + (11.7 - 8.56i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.30894794748112587350517837742, −12.89898015233944767064987974002, −11.60427594353841461743759208640, −10.72018620809571581705048817077, −9.702980184876905157808586236387, −8.257519262789573092788179930675, −7.09495845721774535749960855822, −6.56034268259339721150716358795, −4.16261200371078069099566346600, −1.38020793426979339857986285452, 2.32975223084196899587987655927, 4.59731210162157357292827215192, 6.24198679487060213928870344415, 7.85790097330696964838603901585, 8.948806811357738106736226989401, 9.708102567222294558858892579773, 11.08592332785430726159287620253, 11.86079562398364707054870386361, 12.80957051189579952148733616244, 14.80227095421518121335952113384

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