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Properties

Label 1-89-89.2-r0-0-0
Degree $1$
Conductor $89$
Sign $0.775 - 0.631i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
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 + (0.415 + 0.909i)3-s + (0.415 − 0.909i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.142 − 0.989i)7-s + (−0.142 − 0.989i)8-s + (−0.654 + 0.755i)9-s + (−0.654 − 0.755i)10-s + (−0.142 + 0.989i)11-s + 12-s + (0.415 + 0.909i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.415 + 0.909i)3-s + (0.415 − 0.909i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.142 − 0.989i)7-s + (−0.142 − 0.989i)8-s + (−0.654 + 0.755i)9-s + (−0.654 − 0.755i)10-s + (−0.142 + 0.989i)11-s + 12-s + (0.415 + 0.909i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.522871816 - 0.5416026516i\)
\(L(\frac12)\) \(\approx\) \(1.522871816 - 0.5416026516i\)
\(L(1)\) \(\approx\) \(1.573032251 - 0.3870026379i\)
\(L(1)\) \(\approx\) \(1.573032251 - 0.3870026379i\)

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 \( 1 + (0.415 + 0.909i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
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

−30.56457144750757373555226965367, −30.05949475296484151040406720437, −29.012885660851385095265739134830, −27.2131257187392893281136154852, −25.856336273186526754960138705558, −25.39057214347589092319079790920, −24.27773452141582417624454011374, −23.3203551629105724438031734566, −22.315168963639380999295881338002, −21.349074066301188615740505226650, −19.89918750224695029851928049283, −18.61572332899239849606501780591, −17.93957647870715201870478443103, −16.2151026301776966184315860075, −15.04459581819790221061602672449, −14.27367184353544292079299089366, −13.15083919503864295972099676617, −12.128647585381359288846855963920, −10.98894886844853365112687140896, −8.72404205776212638421443608136, −7.72775376766791358333090129246, −6.481892601591672157740271211753, −5.63386337850225323284484457447, −3.32250497984347422614692200992, −2.583745643055996306517613431177, 1.819430275970310970099015295547, 3.89489489747632466677487595748, 4.32984984599852253575633989533, 5.788992860401070064810797535457, 7.74732261788308338768706419614, 9.44784083552495407322454655658, 10.24067776945653671664890991742, 11.578892088283929229886797831314, 12.866116350489884143444920994652, 13.893617661925020458955047583422, 14.95681002299018345723985416429, 16.136358275005306265162267667596, 16.99175697729263543551434483903, 19.16767832904931851492919313922, 20.15596517832002593529431919677, 20.77450724701350928454744402724, 21.60009012708472353023280979125, 23.08129136789526570405080043334, 23.64930913085092871670089929095, 25.11543142013999247411045482111, 26.16078282446209630298337324186, 27.68164089195683785054291619818, 28.20065177297746388820239911441, 29.402764954016254969132872025429, 30.66651674074149562459295354238

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