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Properties

Label 1-89-89.38-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.314 + 0.949i$
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  + (−0.142 + 0.989i)2-s + (−0.877 − 0.479i)3-s + (−0.959 − 0.281i)4-s + (−0.909 − 0.415i)5-s + (0.599 − 0.800i)6-s + (0.936 − 0.349i)7-s + (0.415 − 0.909i)8-s + (0.540 + 0.841i)9-s + (0.540 − 0.841i)10-s + (−0.415 − 0.909i)11-s + (0.707 + 0.707i)12-s + (−0.479 + 0.877i)13-s + (0.212 + 0.977i)14-s + (0.599 + 0.800i)15-s + (0.841 + 0.540i)16-s + (−0.989 + 0.142i)17-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.877 − 0.479i)3-s + (−0.959 − 0.281i)4-s + (−0.909 − 0.415i)5-s + (0.599 − 0.800i)6-s + (0.936 − 0.349i)7-s + (0.415 − 0.909i)8-s + (0.540 + 0.841i)9-s + (0.540 − 0.841i)10-s + (−0.415 − 0.909i)11-s + (0.707 + 0.707i)12-s + (−0.479 + 0.877i)13-s + (0.212 + 0.977i)14-s + (0.599 + 0.800i)15-s + (0.841 + 0.540i)16-s + (−0.989 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.314 + 0.949i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.314 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3622480504 + 0.5016796543i\)
\(L(\frac12)\) \(\approx\) \(0.3622480504 + 0.5016796543i\)
\(L(1)\) \(\approx\) \(0.5659105210 + 0.2058649345i\)
\(L(1)\) \(\approx\) \(0.5659105210 + 0.2058649345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (-0.877 - 0.479i)T \)
5 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (0.936 - 0.349i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.479 + 0.877i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (-0.212 + 0.977i)T \)
23 \( 1 + (0.977 + 0.212i)T \)
29 \( 1 + (0.349 + 0.936i)T \)
31 \( 1 + (0.977 - 0.212i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.479 + 0.877i)T \)
43 \( 1 + (0.349 - 0.936i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (0.877 - 0.479i)T \)
61 \( 1 + (0.997 - 0.0713i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.909 - 0.415i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.540 + 0.841i)T \)
83 \( 1 + (-0.599 + 0.800i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−30.00160573574821803512063179038, −28.64337507607385184136386828330, −27.93759231806812940130962148134, −27.14959918026148722664857857955, −26.36499251017509872781280630107, −24.34935573573603653671363008350, −23.08562991664038453301946814804, −22.54160781200822592007874665420, −21.375318960173613116061670639493, −20.43351792239674254968555553278, −19.25772381055658863637020158910, −17.85749558322061626013851116487, −17.52481776261290012199689362955, −15.6097047429511724502779521994, −14.82948976512577306628657103916, −12.89808562464358568850017013684, −11.84978640159020520953884835277, −11.080905511876030498783017520986, −10.182108570441094051585991504240, −8.65179188573882496309541689039, −7.23393304302823320620949914933, −5.07249289417163253010005646264, −4.3240482332603530419563868950, −2.570574099345257592197792613661, −0.42450247693666845046464310423, 1.10078580535501323015628029375, 4.3019863533095954167971147661, 5.19439994126594826895034694221, 6.67886724097108587974527829579, 7.740848921107000607610226785339, 8.67352743171861177313363765693, 10.6480660339018926114460675301, 11.71651315190584091946560434287, 13.036450163820425324935038517116, 14.225393219256714734461698110755, 15.60433160176613266673056876150, 16.59617045011522926578612320712, 17.323957277088161678089620896905, 18.57127964401670474145140196912, 19.38206235525908973034244949630, 21.20866987031282066972802725420, 22.49877743740774700709745173327, 23.66995470997430826829221873219, 24.00864639957128242816803411639, 24.870704295354126599994256026252, 26.73250473542954988846661802525, 27.19620020911243276916959052359, 28.27699141177248176210514048915, 29.35415651043178122202651181869, 30.92625477962938825710407317490

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