Properties

Label 1-89-89.70-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.182 - 0.983i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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 + (0.800 − 0.599i)3-s + (−0.142 − 0.989i)4-s + (0.540 − 0.841i)5-s + (−0.0713 + 0.997i)6-s + (−0.212 − 0.977i)7-s + (0.841 + 0.540i)8-s + (0.281 − 0.959i)9-s + (0.281 + 0.959i)10-s + (−0.841 + 0.540i)11-s + (−0.707 − 0.707i)12-s + (−0.599 − 0.800i)13-s + (0.877 + 0.479i)14-s + (−0.0713 − 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (0.800 − 0.599i)3-s + (−0.142 − 0.989i)4-s + (0.540 − 0.841i)5-s + (−0.0713 + 0.997i)6-s + (−0.212 − 0.977i)7-s + (0.841 + 0.540i)8-s + (0.281 − 0.959i)9-s + (0.281 + 0.959i)10-s + (−0.841 + 0.540i)11-s + (−0.707 − 0.707i)12-s + (−0.599 − 0.800i)13-s + (0.877 + 0.479i)14-s + (−0.0713 − 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8035023608 - 0.9660466464i\)
\(L(\frac12)\) \(\approx\) \(0.8035023608 - 0.9660466464i\)
\(L(1)\) \(\approx\) \(0.9148053658 - 0.2780762479i\)
\(L(1)\) \(\approx\) \(0.9148053658 - 0.2780762479i\)

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 \( 1 + (0.800 - 0.599i)T \)
5 \( 1 + (0.540 - 0.841i)T \)
7 \( 1 + (-0.212 - 0.977i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (-0.599 - 0.800i)T \)
17 \( 1 + (-0.755 + 0.654i)T \)
19 \( 1 + (-0.877 + 0.479i)T \)
23 \( 1 + (0.479 + 0.877i)T \)
29 \( 1 + (0.977 - 0.212i)T \)
31 \( 1 + (0.479 - 0.877i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.599 - 0.800i)T \)
43 \( 1 + (0.977 + 0.212i)T \)
47 \( 1 + (-0.989 + 0.142i)T \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (-0.800 - 0.599i)T \)
61 \( 1 + (0.936 - 0.349i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.540 - 0.841i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (-0.281 - 0.959i)T \)
83 \( 1 + (0.0713 - 0.997i)T \)
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

−30.68552790821105923265129854375, −29.29313061103912045068209427229, −28.590723897978487685514105806557, −27.19343227216274349525933915900, −26.46306276687793545210069242654, −25.69804914285747585818444029834, −24.7192592008210579555214951175, −22.55519049192061944643047531434, −21.523996307087088130560817482413, −21.24537615605069912796660349045, −19.654016850652107870849315386565, −18.86859772354948488037246410355, −17.99623754356996427037171185433, −16.451523613850737211636325788795, −15.32703633007801363468260106552, −14.03780398013912990744513762083, −12.91380376169940591278784575652, −11.304486873877405721125837519488, −10.30719812210233918825334928841, −9.2791180083174310178241827152, −8.39085934183671507471711460869, −6.79192552809507817670030348404, −4.69883334568798661375774321044, −2.830564871206753907732120396419, −2.40355637160477953677864645242, 0.62755133181194289322458466678, 2.12447549035521357354775518880, 4.47055827734584369235429098984, 6.07076147473105896187849635021, 7.42543414458771602209818969114, 8.24893804505182216261071779820, 9.506865278147114679650880787547, 10.433820979820747718328110304088, 12.79073328926913945094833034169, 13.45488733197152854945482883880, 14.71183138259170256048274305883, 15.81285231768764980111370864688, 17.27850603937110849983748446927, 17.74118975136775136219014226507, 19.30608262990898058486319706340, 20.031981237648302872729136075120, 21.00940902488127564659544805442, 23.09254377237762692188060589823, 23.935177264429123217806149224362, 24.86303148959643885429632853182, 25.704105524793282767674583354824, 26.50182679973792978772256605666, 27.695144840037103799696062206294, 29.010180325087663929581901506222, 29.66942115219438562922843619232

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