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Properties

Label 2-88-8.5-c5-0-1
Degree $2$
Conductor $88$
Sign $-0.213 + 0.976i$
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  + (−5.64 − 0.405i)2-s + 23.7i·3-s + (31.6 + 4.57i)4-s + 29.2i·5-s + (9.63 − 134. i)6-s − 155.·7-s + (−176. − 38.6i)8-s − 322.·9-s + (11.8 − 165. i)10-s + 121i·11-s + (−108. + 753. i)12-s + 248. i·13-s + (876. + 62.9i)14-s − 695.·15-s + (982. + 289. i)16-s − 234.·17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0716i)2-s + 1.52i·3-s + (0.989 + 0.142i)4-s + 0.523i·5-s + (0.109 − 1.52i)6-s − 1.19·7-s + (−0.976 − 0.213i)8-s − 1.32·9-s + (0.0374 − 0.521i)10-s + 0.301i·11-s + (−0.217 + 1.50i)12-s + 0.408i·13-s + (1.19 + 0.0858i)14-s − 0.798·15-s + (0.959 + 0.282i)16-s − 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.213 + 0.976i$
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: \((2,\ 88,\ (\ :5/2),\ -0.213 + 0.976i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.112558 - 0.139799i\)
\(L(\frac12)\) \(\approx\) \(0.112558 - 0.139799i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.64 + 0.405i)T \)
11 \( 1 - 121iT \)
good3 \( 1 - 23.7iT - 243T^{2} \)
5 \( 1 - 29.2iT - 3.12e3T^{2} \)
7 \( 1 + 155.T + 1.68e4T^{2} \)
13 \( 1 - 248. iT - 3.71e5T^{2} \)
17 \( 1 + 234.T + 1.41e6T^{2} \)
19 \( 1 + 537. iT - 2.47e6T^{2} \)
23 \( 1 + 1.77e3T + 6.43e6T^{2} \)
29 \( 1 + 3.14e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.16e3T + 2.86e7T^{2} \)
37 \( 1 + 9.57e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.06e4T + 1.15e8T^{2} \)
43 \( 1 + 1.27e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.14e4T + 2.29e8T^{2} \)
53 \( 1 - 2.52e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.33e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.32e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.49e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.08e3T + 1.80e9T^{2} \)
73 \( 1 + 7.73e4T + 2.07e9T^{2} \)
79 \( 1 + 5.89e4T + 3.07e9T^{2} \)
83 \( 1 - 3.65e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.94e4T + 5.58e9T^{2} \)
97 \( 1 + 8.92e4T + 8.58e9T^{2} \)
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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

−14.38006219430255537996022575844, −12.63340014263126156674120449357, −11.26128831161866821306132294716, −10.42289754625459396374703360932, −9.632128159330083641271100664368, −8.938532631378504766941278159508, −7.20638238156506812621274220151, −5.95696947264406431952132888983, −3.99082351744516919901414759168, −2.72715241092391025494929696244, 0.10420057954765240431349430750, 1.33106638201342834457798704367, 2.90568883747326435092628020194, 5.91868295855903296820435342949, 6.77570504959833848202116659115, 7.84892614086290895390307856268, 8.840067561039913647671310417768, 10.03296669900968250819241611523, 11.45463774497052499109814862928, 12.59579867187225732850962575394

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