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Properties

Label 1-89-89.56-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.848 + 0.529i$
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  + (−0.959 + 0.281i)2-s + (−0.977 + 0.212i)3-s + (0.841 − 0.540i)4-s + (−0.755 − 0.654i)5-s + (0.877 − 0.479i)6-s + (−0.0713 − 0.997i)7-s + (−0.654 + 0.755i)8-s + (0.909 − 0.415i)9-s + (0.909 + 0.415i)10-s + (0.654 + 0.755i)11-s + (−0.707 + 0.707i)12-s + (−0.212 − 0.977i)13-s + (0.349 + 0.936i)14-s + (0.877 + 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.977 + 0.212i)3-s + (0.841 − 0.540i)4-s + (−0.755 − 0.654i)5-s + (0.877 − 0.479i)6-s + (−0.0713 − 0.997i)7-s + (−0.654 + 0.755i)8-s + (0.909 − 0.415i)9-s + (0.909 + 0.415i)10-s + (0.654 + 0.755i)11-s + (−0.707 + 0.707i)12-s + (−0.212 − 0.977i)13-s + (0.349 + 0.936i)14-s + (0.877 + 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.848 + 0.529i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008406988493 + 0.02933968357i\)
\(L(\frac12)\) \(\approx\) \(0.008406988493 + 0.02933968357i\)
\(L(1)\) \(\approx\) \(0.3886846172 + 0.02571157685i\)
\(L(1)\) \(\approx\) \(0.3886846172 + 0.02571157685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.977 + 0.212i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (-0.0713 - 0.997i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (-0.212 - 0.977i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (-0.349 + 0.936i)T \)
23 \( 1 + (-0.936 - 0.349i)T \)
29 \( 1 + (-0.997 + 0.0713i)T \)
31 \( 1 + (-0.936 + 0.349i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.212 - 0.977i)T \)
43 \( 1 + (-0.997 - 0.0713i)T \)
47 \( 1 + (0.540 + 0.841i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (0.977 + 0.212i)T \)
61 \( 1 + (0.599 + 0.800i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.755 - 0.654i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (-0.877 + 0.479i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−29.78086529480620478729103129835, −28.46073155460101361579813154366, −27.93809413871664694736519012944, −26.911579527900479579688640417554, −25.88962922335204359973859803988, −24.47775976525096943399208554471, −23.67284912936445500345584094325, −21.88417813592971424475471436297, −21.77072408326699900197465653598, −19.6374755147270618748529984671, −18.92146466335947916552515768803, −18.16098146694335030640772526790, −16.88658520150889450567342115374, −16.01005457359672803906464481594, −14.86727741872399096305408448846, −12.69857467786038488273152762804, −11.54373187675026232730648804092, −11.20834152798496649318154472445, −9.65478290884369131057841666735, −8.3011747202760283153130972627, −6.94880819326144988692865146784, −6.000195300900770178086311453332, −3.811245118619601529786373831906, −1.98803201699936633884042758855, −0.024691728035828855224792067908, 1.19958432457352569707188104155, 4.01418104515317284102358564842, 5.4422622998523961762136471113, 6.975521372125305263938846025158, 7.85458085989118686776617859508, 9.53242529246640178261645003739, 10.474995283896960305696689056006, 11.64516051678016991324760946461, 12.61863763403292459157032834580, 14.71229783914827748691828679319, 15.94223762766433572577520013199, 16.70713665502911835698700125916, 17.46946065406744639761793992916, 18.6556549397371757252946873349, 20.120167670907669584069239524905, 20.5474376149442398419471747181, 22.557235574286015826846270175626, 23.397779801354477811184531582923, 24.31821907015321298110722144161, 25.46169197449475742655599898649, 27.06368967863747439109046424533, 27.395417093865847830897587809950, 28.30568332423153626897852297386, 29.434075644386826909040220595957, 30.21310442970779155683428319063

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