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Properties

Label 1-89-89.74-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.0551 - 0.998i$
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  + (0.841 − 0.540i)2-s + (−0.349 + 0.936i)3-s + (0.415 − 0.909i)4-s + (−0.989 + 0.142i)5-s + (0.212 + 0.977i)6-s + (0.599 − 0.800i)7-s + (−0.142 − 0.989i)8-s + (−0.755 − 0.654i)9-s + (−0.755 + 0.654i)10-s + (0.142 − 0.989i)11-s + (0.707 + 0.707i)12-s + (0.936 + 0.349i)13-s + (0.0713 − 0.997i)14-s + (0.212 − 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.349 + 0.936i)3-s + (0.415 − 0.909i)4-s + (−0.989 + 0.142i)5-s + (0.212 + 0.977i)6-s + (0.599 − 0.800i)7-s + (−0.142 − 0.989i)8-s + (−0.755 − 0.654i)9-s + (−0.755 + 0.654i)10-s + (0.142 − 0.989i)11-s + (0.707 + 0.707i)12-s + (0.936 + 0.349i)13-s + (0.0713 − 0.997i)14-s + (0.212 − 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.0551 - 0.998i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.0551 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300382607 - 1.374210936i\)
\(L(\frac12)\) \(\approx\) \(1.300382607 - 1.374210936i\)
\(L(1)\) \(\approx\) \(1.266652511 - 0.4905433802i\)
\(L(1)\) \(\approx\) \(1.266652511 - 0.4905433802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.349 + 0.936i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (0.599 - 0.800i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.936 + 0.349i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.0713 - 0.997i)T \)
23 \( 1 + (-0.997 + 0.0713i)T \)
29 \( 1 + (0.800 + 0.599i)T \)
31 \( 1 + (-0.997 - 0.0713i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.936 + 0.349i)T \)
43 \( 1 + (0.800 - 0.599i)T \)
47 \( 1 + (-0.909 - 0.415i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (0.349 + 0.936i)T \)
61 \( 1 + (0.479 + 0.877i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.989 + 0.142i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.755 - 0.654i)T \)
83 \( 1 + (-0.212 - 0.977i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−30.843198912533427108132791837090, −29.946897484110978911765463773818, −28.39945859451617189426299469295, −27.59945490700295614141262823194, −25.840484356304422255501852251233, −25.00970896690413022768915356854, −24.03667903258288413934653674766, −23.25388917695810596610105942264, −22.50896750554589196079685656648, −21.037927097551266908127612671057, −19.90054215823301985209499302340, −18.52420201000989723100483036047, −17.52811185131923834986562666290, −16.23951799150181024423632786305, −15.15983223056027711322093047910, −14.163474369781910270448634021688, −12.54159653739883030530543409786, −12.215319591833098894288720432788, −11.042353013402073249593873978965, −8.34499751853936281926858299439, −7.78880624670093316631838058525, −6.372043903072021862636370958195, −5.24725523334654671358196993171, −3.75909468264822572246090854148, −1.89949943179092032961985564979, 0.72250587472208156943873799142, 3.30368536391953340022583573492, 4.12016477877857523457942753642, 5.26806313420875469446354636961, 6.82398268855402154300093779434, 8.6400997886029267519899214029, 10.32332515421681253498576107346, 11.24232142260236975925604637891, 11.78951643795591732925862921105, 13.65708406147943743571365359411, 14.55594931870562349682236969972, 15.79196901974568738146300677379, 16.47001832497918573358993620419, 18.29619521610847665471609095420, 19.70129702581244702012958849482, 20.522735556637392291436072926254, 21.48284842500142991913564091912, 22.51225049360319751485366859643, 23.53290087803525590832027388246, 24.039643773102572176819035527004, 26.01003437355449877711078715993, 27.272736201077563263123746837, 27.744007293090961174894179995872, 29.02593929131797484916249619964, 30.13230121704679181802795553845

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