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Properties

Label 2-88-88.53-c1-0-5
Degree $2$
Conductor $88$
Sign $0.959 - 0.283i$
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  + (1.40 + 0.116i)2-s + (0.317 + 0.437i)3-s + (1.97 + 0.329i)4-s + (−2.75 + 0.896i)5-s + (0.396 + 0.653i)6-s + (−2.10 − 1.53i)7-s + (2.74 + 0.694i)8-s + (0.836 − 2.57i)9-s + (−3.99 + 0.941i)10-s + (−2.98 − 1.45i)11-s + (0.482 + 0.967i)12-s + (2.48 + 0.808i)13-s + (−2.79 − 2.40i)14-s + (−1.26 − 0.921i)15-s + (3.78 + 1.29i)16-s + (2.12 + 6.55i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0825i)2-s + (0.183 + 0.252i)3-s + (0.986 + 0.164i)4-s + (−1.23 + 0.400i)5-s + (0.161 + 0.266i)6-s + (−0.796 − 0.578i)7-s + (0.969 + 0.245i)8-s + (0.278 − 0.858i)9-s + (−1.26 + 0.297i)10-s + (−0.899 − 0.437i)11-s + (0.139 + 0.279i)12-s + (0.690 + 0.224i)13-s + (−0.745 − 0.642i)14-s + (−0.327 − 0.237i)15-s + (0.945 + 0.324i)16-s + (0.516 + 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.959 - 0.283i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.959 - 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43290 + 0.207093i\)
\(L(\frac12)\) \(\approx\) \(1.43290 + 0.207093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.116i)T \)
11 \( 1 + (2.98 + 1.45i)T \)
good3 \( 1 + (-0.317 - 0.437i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (2.75 - 0.896i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.10 + 1.53i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2.48 - 0.808i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.12 - 6.55i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.940 + 1.29i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.01T + 23T^{2} \)
29 \( 1 + (1.17 - 1.61i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.60 - 4.92i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.04 + 6.94i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.03 - 2.20i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.47iT - 43T^{2} \)
47 \( 1 + (-9.58 + 6.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.23 + 0.399i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.206 - 0.284i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.93 + 0.952i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + (-1.07 - 3.30i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.07 + 0.784i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.33 + 4.11i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (15.2 - 4.93i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (-2.66 + 8.19i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.31985199625312764961090124039, −13.07743543286210162900056460392, −12.32009791784812900211039189795, −11.08918210244766779008310799812, −10.25822399850255405615740760833, −8.324704305125060691281770981764, −7.16414897909523025248782183018, −6.03885034751233470425979452261, −4.00257225118158679874891905106, −3.42284998376609541776402455065, 2.76358101405250224920510100493, 4.31843202039604297818197377215, 5.63653910628456814422894374462, 7.30354162654600205949070701483, 8.101841591328025546939940898704, 9.955140818326837247732841466593, 11.28671447710037715965521122376, 12.22550247954250914712898396221, 12.99435881998535780961590875503, 13.88733340968048798810585809293

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