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Properties

Label 1-89-89.39-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.514 + 0.857i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
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  + (−0.654 + 0.755i)2-s + (−0.142 + 0.989i)3-s + (−0.142 − 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (0.841 + 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.959 + 0.281i)10-s + (0.841 − 0.540i)11-s + 12-s + (−0.142 + 0.989i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.654 − 0.755i)17-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.142 + 0.989i)3-s + (−0.142 − 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (0.841 + 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.959 + 0.281i)10-s + (0.841 − 0.540i)11-s + 12-s + (−0.142 + 0.989i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.654 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3951433559 + 0.6978543921i\)
\(L(\frac12)\) \(\approx\) \(0.3951433559 + 0.6978543921i\)
\(L(1)\) \(\approx\) \(0.6400081919 + 0.5598040086i\)
\(L(1)\) \(\approx\) \(0.6400081919 + 0.5598040086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−30.127422363828045085271367144428, −29.22018741242141992204026999558, −28.18412419858423829509506855430, −27.38422954358177506462372180007, −25.805277180374647418484910824891, −25.06885498086698112963905497989, −24.0199815639355367904679145810, −22.59958252405442202438538762909, −21.42263787205243070114696077125, −20.21577661426071455238829037878, −19.63642065915735715476383418846, −17.988237999505937750703132520664, −17.56957239437240121468648960311, −16.769344960973824628914540562396, −14.48742274077587775193504136833, −13.24031503012935915340613117088, −12.49377513716920543194864971735, −11.25832161119150261033615421790, −10.06998263371788748025492894407, −8.63116976299735651869682208101, −7.717316431084689291356166250787, −6.19911828284714245031632187741, −4.37789990188251279468999178588, −2.22110941692914854171145366191, −1.25331612305958951063505042462, 2.13624956024163401209057930668, 4.4527724706586027268345517501, 5.704029443126340233885692993289, 6.740241878374268276079306102590, 8.68216215638373646317487514158, 9.31552063556647749478886888877, 10.6186752454322957280072367682, 11.509994222789983003038931350494, 14.103467283321704421522161670216, 14.49281801232420803652178575181, 15.74218046042242202022705639565, 16.8660690054886048997148494041, 17.690120752625339585530291906908, 18.73557827629849685366316393967, 20.15403803970745553174837512253, 21.58063627690250508030638532584, 22.1458167746950067788754107736, 23.652390559436485296159479219462, 24.81863626757689901701826632022, 25.74604274494142372756488840610, 26.74207953760773639029943832622, 27.46877360977858570674873776872, 28.491572422337821761341260684206, 29.47332988694783693921840298540, 31.11829796609777664244250768633

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