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Properties

Label 1-87-87.80-r1-0-0
Degree $1$
Conductor $87$
Sign $0.995 - 0.0915i$
Analytic cond. $9.34944$
Root an. cond. $9.34944$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.900 − 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.222 + 0.974i)8-s + (−0.623 + 0.781i)10-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.623 + 0.781i)19-s + (0.222 − 0.974i)20-s + (0.623 + 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.900 − 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.222 + 0.974i)8-s + (−0.623 + 0.781i)10-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.623 + 0.781i)19-s + (0.222 − 0.974i)20-s + (0.623 + 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.995 - 0.0915i$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ 0.995 - 0.0915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381967654 - 0.06338362945i\)
\(L(\frac12)\) \(\approx\) \(1.381967654 - 0.06338362945i\)
\(L(1)\) \(\approx\) \(0.9516766977 + 0.04314270546i\)
\(L(1)\) \(\approx\) \(0.9516766977 + 0.04314270546i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.623 + 0.781i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.222 - 0.974i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−30.21748263279678634787048497774, −29.27477651256111997424522265857, −28.351286611600704243432724474385, −27.21846788406926130503325077831, −26.17993095429844728574745818395, −25.5356715651914600464753490073, −24.24402833593110339256779142836, −22.824344744603054851120563695001, −21.333453857157457433613679271157, −20.86839856609466836804934408248, −19.524434948673595744753411483861, −18.40488681376623509024816076256, −17.40596155558616381837933303088, −16.778828668132954310437895031367, −15.04261261199718345646323284611, −13.81254191723627236066348289225, −12.45059783154426658717023058653, −11.04840894672485534753698764763, −10.16964635649284252122431691745, −9.14245120538769304615165158096, −7.57283617801599922167835136104, −6.62157771820450102087838602181, −4.55043396554692288353333910642, −2.63844840022674518823456759839, −1.34387434800209980896321237678, 1.03919919180631091417907524243, 2.565678035753751254570614190897, 5.34763413824134115769102913493, 5.966206671137289871073379983422, 7.85740449260935651522557533537, 8.7454463331391463698612262088, 9.91406544201559092567263713727, 11.05587089209457491380485186994, 12.54293014860853623615879166214, 14.1032550030225855659362540029, 15.14230741421501729979347680591, 16.42832111316412115808242348913, 17.36778646807360011377720163223, 18.30648688046024662462300907219, 19.27884084835661047533018562161, 20.80929928233125871292546302158, 21.426097988289480280623327948909, 23.16181947138156159859449809486, 24.61191577411862892109831718138, 24.92252852225493202415397396881, 26.05001914186383666965591255409, 27.38109988476432575669036917173, 28.00967049031699070646403851865, 29.22398847987132682715029259893, 29.89589946739803501439302954641

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