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Properties

Label 1-88-88.59-r1-0-0
Degree $1$
Conductor $88$
Sign $-0.286 - 0.958i$
Analytic cond. $9.45691$
Root an. cond. $9.45691$
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

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

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.286 - 0.958i$
Analytic conductor: \(9.45691\)
Root analytic conductor: \(9.45691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (1:\ ),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5379672705 - 0.7222362654i\)
\(L(\frac12)\) \(\approx\) \(0.5379672705 - 0.7222362654i\)
\(L(1)\) \(\approx\) \(0.7544201683 - 0.2254452602i\)
\(L(1)\) \(\approx\) \(0.7544201683 - 0.2254452602i\)

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

−30.749049288467832805811965833300, −29.27058989329998240294421071537, −28.32242011369709066519646591276, −27.7060162870340339058512820666, −26.7587561200385200324488019450, −25.24645847054428223901972314260, −23.92889141244392233077568412562, −23.516125664946322230304971862492, −21.77498525093982904737272076387, −21.31945331335786340483824991135, −20.09501667301295572888085489169, −18.64727375899107117811179121548, −17.37087588201107213450517135859, −16.57875295019115192410832754313, −15.5239456843187996220246169586, −14.38070898743556597492460844903, −12.47477288389112366776316117739, −11.88220812089565766902717285000, −10.60501969387131997435116337733, −9.20658259786213579189227322050, −8.12380395916241158045170991383, −6.21846396710851520121857741438, −5.00204012520231684109689979898, −4.07661181760113763867905255468, −1.54908070691974308815909262611, 0.48039704911832046141342113778, 2.42006020731366506099652799193, 4.375267203339177563519492033082, 5.84619498761786799930468828347, 7.188318316037918327520181603262, 7.93883490845675378442245332401, 10.20228397442539215374316570459, 11.073631809210209041022904154547, 12.01852821742871836922482544763, 13.457200432559701110977975652667, 14.555518108344083399394838360463, 15.85561384648880259102292210610, 17.31049473321381449245979693407, 17.9779632035732095276997816909, 19.02804251005549747228003278223, 20.23044707444550582907302132238, 21.75031347072144199337214722873, 22.73383990359546690981187964629, 23.51900252688907124016888526589, 24.51272350827212491144800694670, 25.76318759561328284788759482390, 27.16185103480194576044785034307, 27.70655267128661182189185200892, 29.206587494723186983995216007898, 30.157471266760955339019509104429

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