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Properties

Label 1-89-89.46-r1-0-0
Degree $1$
Conductor $89$
Sign $0.743 + 0.669i$
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 + (0.479 − 0.877i)3-s + (−0.959 − 0.281i)4-s + (0.909 + 0.415i)5-s + (0.800 + 0.599i)6-s + (0.349 + 0.936i)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 + (0.877 + 0.479i)13-s + (−0.977 + 0.212i)14-s + (0.800 − 0.599i)15-s + (0.841 + 0.540i)16-s + (0.989 − 0.142i)17-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (0.479 − 0.877i)3-s + (−0.959 − 0.281i)4-s + (0.909 + 0.415i)5-s + (0.800 + 0.599i)6-s + (0.349 + 0.936i)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 + (0.877 + 0.479i)13-s + (−0.977 + 0.212i)14-s + (0.800 − 0.599i)15-s + (0.841 + 0.540i)16-s + (0.989 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.743 + 0.669i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.743 + 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.917130953 + 0.7359954346i\)
\(L(\frac12)\) \(\approx\) \(1.917130953 + 0.7359954346i\)
\(L(1)\) \(\approx\) \(1.295936163 + 0.3768634466i\)
\(L(1)\) \(\approx\) \(1.295936163 + 0.3768634466i\)

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 \( 1 + (0.479 - 0.877i)T \)
5 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (0.349 + 0.936i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (0.877 + 0.479i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (0.977 + 0.212i)T \)
23 \( 1 + (-0.212 + 0.977i)T \)
29 \( 1 + (0.936 - 0.349i)T \)
31 \( 1 + (-0.212 - 0.977i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.877 + 0.479i)T \)
43 \( 1 + (0.936 + 0.349i)T \)
47 \( 1 + (0.281 - 0.959i)T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (-0.479 - 0.877i)T \)
61 \( 1 + (0.0713 + 0.997i)T \)
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 \( 1 + (-0.800 - 0.599i)T \)
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

−30.29079234020570132414459297524, −28.90165135795852269273657441301, −28.143709899160777071655632078002, −27.1393990406695431438027110660, −26.11770446006453020236829138754, −25.30244337637121879478059987549, −23.44229316473786883654877505648, −22.390793841498443832649453793683, −21.18530433087096349323804002614, −20.58979477578021908013157806291, −19.95980381749484494070309789232, −18.24485524295940447147240704917, −17.311667228704688824121550718171, −16.16945588118465628244226811022, −14.37890466159056811260030990756, −13.68619245164497713559225630633, −12.47010918355830191495836484660, −10.67778422389887492775226557260, −10.17567030999640027031432782217, −9.061101663314365362696015670, −7.84888226363747921093409770033, −5.33126997264259611601808539630, −4.296474653074909375165035150305, −2.880429540744238184535205455490, −1.27397238038732689406135095447, 1.36074329362082791131536492567, 3.12816161830662613504142931817, 5.57552559000400205622004321541, 6.20012487235712350344155659665, 7.66957196744606758773469641655, 8.676308508035221244787308293936, 9.740937894896860746666384189410, 11.661214310339753826995545752259, 13.30205533344789626753404462177, 13.92612921900149011816391424323, 14.9154444582714920148383021006, 16.24216101862248706998526301117, 17.65748288450779069089121380924, 18.47270600708047497674810120084, 18.97304011557003591357704090046, 20.94992075121350240181772359351, 21.96901712948705413376142837462, 23.35305657593734884435488280811, 24.290222354798723388088823465794, 25.223705421073165519691357473256, 25.77664673804168639163264412687, 26.85714003622456731428462292780, 28.32449274544669823894342945596, 29.3251480866350589867801624110, 30.617676221274975482255715395714

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