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Properties

Label 2-88-8.5-c5-0-20
Degree $2$
Conductor $88$
Sign $-0.601 - 0.799i$
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  + (5.52 − 1.20i)2-s + 27.7i·3-s + (29.0 − 13.3i)4-s + 87.4i·5-s + (33.4 + 153. i)6-s + 14.3·7-s + (144. − 108. i)8-s − 524.·9-s + (105. + 483. i)10-s + 121i·11-s + (369. + 806. i)12-s − 1.11e3i·13-s + (79.3 − 17.3i)14-s − 2.42e3·15-s + (668. − 775. i)16-s + 621.·17-s + ⋯
L(s)  = 1  + (0.976 − 0.213i)2-s + 1.77i·3-s + (0.908 − 0.416i)4-s + 1.56i·5-s + (0.379 + 1.73i)6-s + 0.110·7-s + (0.799 − 0.601i)8-s − 2.16·9-s + (0.333 + 1.52i)10-s + 0.301i·11-s + (0.740 + 1.61i)12-s − 1.82i·13-s + (0.108 − 0.0236i)14-s − 2.78·15-s + (0.652 − 0.757i)16-s + 0.521·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.601 - 0.799i$
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: \((2,\ 88,\ (\ :5/2),\ -0.601 - 0.799i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.42121 + 2.84734i\)
\(L(\frac12)\) \(\approx\) \(1.42121 + 2.84734i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.52 + 1.20i)T \)
11 \( 1 - 121iT \)
good3 \( 1 - 27.7iT - 243T^{2} \)
5 \( 1 - 87.4iT - 3.12e3T^{2} \)
7 \( 1 - 14.3T + 1.68e4T^{2} \)
13 \( 1 + 1.11e3iT - 3.71e5T^{2} \)
17 \( 1 - 621.T + 1.41e6T^{2} \)
19 \( 1 - 1.62e3iT - 2.47e6T^{2} \)
23 \( 1 - 347.T + 6.43e6T^{2} \)
29 \( 1 - 6.69e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.17e3T + 2.86e7T^{2} \)
37 \( 1 - 2.86e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.04e3T + 1.15e8T^{2} \)
43 \( 1 - 7.98iT - 1.47e8T^{2} \)
47 \( 1 - 1.20e4T + 2.29e8T^{2} \)
53 \( 1 - 3.03e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.54e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.50e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.45e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.67e4T + 1.80e9T^{2} \)
73 \( 1 + 6.69e4T + 2.07e9T^{2} \)
79 \( 1 - 5.08e4T + 3.07e9T^{2} \)
83 \( 1 - 2.73e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.79e4T + 5.58e9T^{2} \)
97 \( 1 - 1.39e5T + 8.58e9T^{2} \)
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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

−14.07741244219839483633177875919, −12.39308815821939827506466164569, −11.01100882751841993775207078926, −10.52983920971765982221677991501, −9.856785279536924089491111052713, −7.77764615229428414226736175769, −6.12081856469558954720890463325, −5.06529690327472957451691854272, −3.56412003365314640074468874806, −2.93450171597297694895497969418, 1.00226201721253096254576755810, 2.22116710447067547731816720419, 4.46048144018899305126159337938, 5.79905640734739380907013857101, 6.87934936454240257387138394435, 7.988302411453230029174596360128, 8.985240515270715793168417272829, 11.62058143174802958725968231402, 11.92783860062008264505474411874, 13.05176408955907850462479680345

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