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Properties

Label 1-116-116.71-r1-0-0
Degree $1$
Conductor $116$
Sign $-0.426 + 0.904i$
Analytic cond. $12.4659$
Root an. cond. $12.4659$
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  + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)5-s + (0.222 + 0.974i)7-s + (−0.900 + 0.433i)9-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)15-s − 17-s + (−0.222 + 0.974i)19-s + (0.900 − 0.433i)21-s + (−0.623 + 0.781i)23-s + (−0.222 + 0.974i)25-s + (0.623 + 0.781i)27-s + (0.623 + 0.781i)31-s + (−0.222 + 0.974i)33-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s + (0.623 + 0.781i)5-s + (0.222 + 0.974i)7-s + (−0.900 + 0.433i)9-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)15-s − 17-s + (−0.222 + 0.974i)19-s + (0.900 − 0.433i)21-s + (−0.623 + 0.781i)23-s + (−0.222 + 0.974i)25-s + (0.623 + 0.781i)27-s + (0.623 + 0.781i)31-s + (−0.222 + 0.974i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(12.4659\)
Root analytic conductor: \(12.4659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (1:\ ),\ -0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3873039803 + 0.6108221531i\)
\(L(\frac12)\) \(\approx\) \(0.3873039803 + 0.6108221531i\)
\(L(1)\) \(\approx\) \(0.8224339029 + 0.06530580250i\)
\(L(1)\) \(\approx\) \(0.8224339029 + 0.06530580250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 - T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (0.222 - 0.974i)T \)
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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

−28.670450857839150842654924231340, −27.727947480250356650004057246396, −26.510256261781013315763539000371, −26.0238326266283711928599179481, −24.45123032641214200793730408515, −23.630504671917693838518861924351, −22.36380729561279343438520349707, −21.423900357808109209230143064127, −20.506049580297490521983198847800, −19.85180707457544453313400135328, −17.863524491756828403824528251930, −17.127529649531434192444406613044, −16.26919365129113640136053806227, −15.123805684107196776863418772636, −13.89495483391586357814567004478, −12.84824524071623783258247662106, −11.37207717124598136105772891602, −10.234787614282266503574194033070, −9.46580841248566995397633101632, −8.14476870070495524247739012355, −6.50740830439138057390810282357, −4.900916703698085481639626191299, −4.41810703650004202168804203825, −2.3942980996734421335088278951, −0.283159292355091296654269910263, 1.96737938381274718064194390829, 2.843966946717243928353220152810, 5.33090073098266117991022035803, 6.14010089430768304555479467147, 7.42510285589409284746536120580, 8.5354377031285045260109921624, 10.10273681801537723278559676354, 11.28154167945405237052882199692, 12.38362278962730028288381008585, 13.433576398381517418918680663522, 14.46341831319853719966123677352, 15.59011750565657448503803662018, 17.20820885060306369007663275485, 18.108766941838880754596271512201, 18.68277501662830601635582947004, 19.80652775286241718543939232144, 21.38230196046972467839456513358, 22.166231947339983334544207763725, 23.214093069861445979663604563124, 24.42636895815991404458889414689, 25.10390949959938582193182803267, 26.08507041246747252223663840778, 27.288864652734442735526766504668, 28.73295093505818622858338716209, 29.228671909149444607300042500839

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