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Properties

Label 2-88-11.5-c1-0-0
Degree $2$
Conductor $88$
Sign $0.830 - 0.557i$
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  + (0.743 + 0.540i)3-s + (−1.24 + 3.82i)5-s + (3.01 − 2.18i)7-s + (−0.665 − 2.04i)9-s + (−1.97 − 2.66i)11-s + (0.324 + 0.998i)13-s + (−2.99 + 2.17i)15-s + (−0.291 + 0.898i)17-s + (−1.92 − 1.40i)19-s + 3.42·21-s − 1.73·23-s + (−9.05 − 6.58i)25-s + (1.46 − 4.50i)27-s + (2.22 − 1.61i)29-s + (1.47 + 4.55i)31-s + ⋯
L(s)  = 1  + (0.429 + 0.311i)3-s + (−0.556 + 1.71i)5-s + (1.13 − 0.827i)7-s + (−0.221 − 0.683i)9-s + (−0.594 − 0.804i)11-s + (0.0899 + 0.276i)13-s + (−0.772 + 0.561i)15-s + (−0.0708 + 0.217i)17-s + (−0.442 − 0.321i)19-s + 0.746·21-s − 0.362·23-s + (−1.81 − 1.31i)25-s + (0.281 − 0.867i)27-s + (0.412 − 0.300i)29-s + (0.265 + 0.817i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.830 - 0.557i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.830 - 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01484 + 0.308867i\)
\(L(\frac12)\) \(\approx\) \(1.01484 + 0.308867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (1.97 + 2.66i)T \)
good3 \( 1 + (-0.743 - 0.540i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.24 - 3.82i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.01 + 2.18i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.324 - 0.998i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.291 - 0.898i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.92 + 1.40i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 + (-2.22 + 1.61i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.47 - 4.55i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.71 - 1.24i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.38 - 5.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.431T + 43T^{2} \)
47 \( 1 + (5.11 + 3.71i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0976 - 0.300i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.54 - 4.75i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.21 - 12.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.68T + 67T^{2} \)
71 \( 1 + (3.97 - 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.84 + 2.06i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.73 + 8.40i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.53 + 13.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.43T + 89T^{2} \)
97 \( 1 + (-0.457 - 1.40i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.43028998381586334170569311219, −13.66621083094034496164483357495, −11.75414983401379221497055885088, −10.95915928713726697599965800460, −10.23497222164232516142418729460, −8.493843603278061094209525825390, −7.51498196767169614456958766816, −6.32512830560935829903076658066, −4.19875310245102798135588155225, −2.96470734760702677030878070680, 1.98093926901652019296961263951, 4.58679618610465481788523530638, 5.38496339913248008121413490930, 7.86049936477135540788034512537, 8.221778214402713924512348670013, 9.306419441742929219979090995095, 11.03914032682247638637835151713, 12.22503073321939028322787194225, 12.80443673488999348341394680836, 13.99502953419827001368641620079

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