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Properties

Label 1-89-89.5-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.987 + 0.159i$
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  + (−0.142 − 0.989i)2-s + (0.281 − 0.959i)3-s + (−0.959 + 0.281i)4-s + (−0.415 − 0.909i)5-s + (−0.989 − 0.142i)6-s + (−0.909 + 0.415i)7-s + (0.415 + 0.909i)8-s + (−0.841 − 0.540i)9-s + (−0.841 + 0.540i)10-s + (0.415 − 0.909i)11-s + i·12-s + (−0.281 + 0.959i)13-s + (0.540 + 0.841i)14-s + (−0.989 + 0.142i)15-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.281 − 0.959i)3-s + (−0.959 + 0.281i)4-s + (−0.415 − 0.909i)5-s + (−0.989 − 0.142i)6-s + (−0.909 + 0.415i)7-s + (0.415 + 0.909i)8-s + (−0.841 − 0.540i)9-s + (−0.841 + 0.540i)10-s + (0.415 − 0.909i)11-s + i·12-s + (−0.281 + 0.959i)13-s + (0.540 + 0.841i)14-s + (−0.989 + 0.142i)15-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.987 + 0.159i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05383507751 - 0.6696497986i\)
\(L(\frac12)\) \(\approx\) \(-0.05383507751 - 0.6696497986i\)
\(L(1)\) \(\approx\) \(0.4227195087 - 0.6563046941i\)
\(L(1)\) \(\approx\) \(0.4227195087 - 0.6563046941i\)

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

−31.3090858208954420525568091651, −30.35792757588385474849119406164, −28.62810249418552062661621683627, −27.44335720829008720123096094335, −26.795946663489885291820501918181, −25.8147582074026657959543468538, −25.24743415715568559227750429249, −23.43709266810166616293925474360, −22.57707969470194018401887819311, −22.069839385559545370173297492, −20.16403074578820235527783044004, −19.30806340842292035864122666229, −17.91697459808745445302612675011, −16.74057951989656903716809616753, −15.74434324856105644639566144388, −14.89583419834210176491023694303, −14.10182981821580323219530379607, −12.49031946882040458613216894588, −10.35122766495378718812714488789, −9.9840938830040114898823954484, −8.39033790042250711390596935699, −7.19216340105938312733499716050, −5.950553517244290084198415412044, −4.31146755830359505048616410833, −3.26076147890326984885611130269, 0.75369311530743708232224567048, 2.46040976411319038751567094288, 3.82362860229860818280797092460, 5.6155841285258051370745636576, 7.38521894960470526963034013514, 8.84984969502282747861356237863, 9.37021857345793047997084626250, 11.56385984543244622563390454084, 12.08036212723490058679766099366, 13.260340301753153145682415010811, 13.99608241707614012840152091630, 16.02867553940116681853659540634, 17.19392213705471971381146750198, 18.53216356118750724942479438825, 19.40597299970850613509882390512, 19.94510098487709822929870835388, 21.26475997627484847754296178639, 22.4533108661668713836654643457, 23.64202929096581668834874846392, 24.565864368120083593817285561033, 25.83134656997317895884346452128, 26.97342604490997771989122392404, 28.261697640420586117815198372033, 29.04600260629535461270164677689, 29.73123962823716945147485425523

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