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Properties

Label 1-88-88.37-r0-0-0
Degree $1$
Conductor $88$
Sign $0.286 + 0.958i$
Analytic cond. $0.408670$
Root an. cond. $0.408670$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s − 21-s + 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)35-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s − 21-s + 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(0.408670\)
Root analytic conductor: \(0.408670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (0:\ ),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8924441364 + 0.6647488628i\)
\(L(\frac12)\) \(\approx\) \(0.8924441364 + 0.6647488628i\)
\(L(1)\) \(\approx\) \(1.070335140 + 0.4416397990i\)
\(L(1)\) \(\approx\) \(1.070335140 + 0.4416397990i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.51429412513386121883114136964, −29.255210021947870510013318570085, −28.53712352388948727405972112080, −26.98040669708874576689223119347, −26.106232807039725285688469647187, −25.067394626625363077491365757790, −24.010945237096412921367904895978, −23.33867631647104698925977857358, −21.64331977010997992429667588101, −20.4303546605954520208120001332, −19.6636378653633855374064291715, −18.85703358573236135433717200181, −17.24625746022053531305578073163, −16.261586481695591093279243231473, −14.97267548479408369099770213601, −13.60268726234987235537203376566, −12.88485344474850482489505257668, −11.75202855783698179707577079018, −9.81122865774917106542866881623, −8.86010485465199390682801119306, −7.64091074716715577527103382974, −6.48038819219605510860352280455, −4.51133607824761180530172099578, −3.175516719878358544055146768572, −1.28552444624621651567010587123, 2.72019739626508544739245282292, 3.40682081031153267409199794373, 5.25115521320241938508874235625, 6.936424135767199980785170805793, 8.15400384848128938660763966813, 9.57894225416485948505504232160, 10.39037058496070173849432723213, 11.88544306057441810066005663096, 13.360606142722183885976574822740, 14.546423466434691579883742230, 15.427248049936141145778838601607, 16.27589943210057285571709482550, 18.109124210828626649525328012920, 19.10376116275445776700145196243, 19.95721440845452326697021484293, 21.226582547745601694180554337345, 22.34416961249357682094333326093, 22.96870876150210591038124817216, 24.9825820647005053867901084134, 25.42602144417262692485651552783, 26.77453164151395179562237479132, 27.21088758345364092026153221085, 28.69003486131820503588897714705, 29.913239163543869937581636170816, 30.96191964039900575574183467361

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