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Properties

Label 2-88-8.5-c5-0-29
Degree $2$
Conductor $88$
Sign $0.232 + 0.972i$
Analytic cond. $14.1137$
Root an. cond. $3.75683$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.43 − 5.10i)2-s + 3.32i·3-s + (−20.1 − 24.8i)4-s + 28.4i·5-s + (16.9 + 8.09i)6-s + 176.·7-s + (−176. + 42.1i)8-s + 231.·9-s + (145. + 69.4i)10-s + 121i·11-s + (82.7 − 66.9i)12-s − 825. i·13-s + (429. − 900. i)14-s − 94.7·15-s + (−213. + 1.00e3i)16-s + 682.·17-s + ⋯
L(s)  = 1  + (0.430 − 0.902i)2-s + 0.213i·3-s + (−0.629 − 0.777i)4-s + 0.509i·5-s + (0.192 + 0.0918i)6-s + 1.36·7-s + (−0.972 + 0.232i)8-s + 0.954·9-s + (0.460 + 0.219i)10-s + 0.301i·11-s + (0.165 − 0.134i)12-s − 1.35i·13-s + (0.585 − 1.22i)14-s − 0.108·15-s + (−0.208 + 0.977i)16-s + 0.572·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(14.1137\)
Root analytic conductor: \(3.75683\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :5/2),\ 0.232 + 0.972i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.97138 - 1.55509i\)
\(L(\frac12)\) \(\approx\) \(1.97138 - 1.55509i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.43 + 5.10i)T \)
11 \( 1 - 121iT \)
good3 \( 1 - 3.32iT - 243T^{2} \)
5 \( 1 - 28.4iT - 3.12e3T^{2} \)
7 \( 1 - 176.T + 1.68e4T^{2} \)
13 \( 1 + 825. iT - 3.71e5T^{2} \)
17 \( 1 - 682.T + 1.41e6T^{2} \)
19 \( 1 + 2.61e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.21e3T + 6.43e6T^{2} \)
29 \( 1 - 2.19e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.63e3T + 2.86e7T^{2} \)
37 \( 1 + 6.59e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.66e4T + 1.15e8T^{2} \)
43 \( 1 + 4.72e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.34e4T + 2.29e8T^{2} \)
53 \( 1 - 2.07e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.45e3iT - 7.14e8T^{2} \)
61 \( 1 - 2.81e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.81e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.78e4T + 1.80e9T^{2} \)
73 \( 1 + 5.95e4T + 2.07e9T^{2} \)
79 \( 1 + 5.31e4T + 3.07e9T^{2} \)
83 \( 1 + 4.34e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.49e4T + 5.58e9T^{2} \)
97 \( 1 + 1.78e5T + 8.58e9T^{2} \)
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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

−12.89242812853531869875258798482, −11.82710871047657338762723557425, −10.74045058730513161615772029010, −10.16002131231903574351938633439, −8.662103341105916397193667022848, −7.23524780010697576278111565475, −5.36085873516009027197818725874, −4.36776398510854031812373047505, −2.73474271006351548822546819217, −1.11832158566203135789488665460, 1.49695238702692411734289892414, 4.07422744798092479321896283475, 5.01991645283969454055289312579, 6.46533902766708990664902895096, 7.77435091409739570497616901251, 8.511646573579186925088565336999, 9.938034775182320363974784227484, 11.70996242692852832345463344678, 12.41218450967780471848918242451, 13.74835962628590039705687219170

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