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Properties

Label 1-89-89.19-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.986 - 0.164i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03790294059 + 0.4568361000i\)
\(L(\frac12)\) \(\approx\) \(-0.03790294059 + 0.4568361000i\)
\(L(1)\) \(\approx\) \(0.4634682498 + 0.3118942765i\)
\(L(1)\) \(\approx\) \(0.4634682498 + 0.3118942765i\)

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

−29.58696292068208802646166196673, −28.9906021906397357089079154811, −27.70403413699389257942322293811, −26.702850081427467547544553550308, −25.79211659520468682899655316680, −24.45358668988028197494456454214, −23.02436147071270223961342342350, −22.32035972167849494381581990049, −21.09024799530024357718534265497, −19.96287479816282901238685184979, −18.55271697257360804452349296599, −18.07145954308936978969096990301, −17.13364707210648086362593492633, −15.9380549590854197860396623369, −13.6896309941218901628595964400, −13.20004861439844039056787940459, −11.47554292834164114001055581914, −10.81411805589796213475384032182, −9.90575137042231719224144657250, −7.92530203004182068629272788826, −7.07601039036145973115700715671, −5.49442981041210163400922575648, −3.45901561023908452071911802445, −1.85417874212092923564249988101, −0.28827835786843627618559874136, 1.69747290602692394123726080964, 4.692287798317420215914581279373, 5.49008169703011934215236581928, 6.603881294166808181962276634297, 8.50163107312386145582312087434, 9.331930453934901931052809771671, 10.458764676201818974128150241524, 11.808080701772172914138255083721, 13.243733096967645386949199017839, 14.90665278379054347403302057815, 15.87612617396172532435206434663, 16.619047501688515487148906972846, 17.84571373924071226319200470329, 18.370973973470688320242233043069, 20.1937210503010618226931856151, 21.2793450918595101721716911109, 22.36825731349651642694602634126, 23.772168922360878290716112448358, 24.37464164226691966329483406676, 25.69046676523427551871146553220, 26.54763797433535834910246410972, 27.960098220224834382795213947586, 28.465727758864261978528495184450, 28.97589013200255119948998305167, 31.10747954850693772223873622617

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