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Properties

Label 1-117-117.5-r0-0-0
Degree $1$
Conductor $117$
Sign $0.451 - 0.892i$
Analytic cond. $0.543345$
Root an. cond. $0.543345$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s − i·8-s − 10-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − i·19-s + (0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s − i·8-s − 10-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − i·19-s + (0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6627454573 - 0.4073647817i\)
\(L(\frac12)\) \(\approx\) \(0.6627454573 - 0.4073647817i\)
\(L(1)\) \(\approx\) \(0.7513119712 - 0.2690818449i\)
\(L(1)\) \(\approx\) \(0.7513119712 - 0.2690818449i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−29.46552854829237482136891631553, −28.2488984374971904272865577673, −27.31334445845932270224741709997, −26.18294477377791905617721341639, −25.37893983083887606455602837819, −24.825664086402162590369519367276, −23.374066480882451813536131945449, −22.27141371440763895466688960106, −21.2194481487548924706199798845, −19.75874593426902082587267154346, −18.877473707306345753074682465715, −18.0624002336997489846223323995, −16.87832007811254966170440817043, −16.1166002105140970783105284812, −14.731563065686271013686220216594, −13.96165510077115066046053637968, −12.33229395291520043914597702819, −10.90731086737305665399784308475, −9.75459644215886208058897997997, −9.1250032826555345334263324745, −7.58830433237937986808229763283, −6.25996672834426855424659501594, −5.701554867275706860090861350932, −3.22778295962396936293697856981, −1.64168339434390891498468414047, 1.1308832970083252363749411723, 2.66905547448053043695961191375, 4.21135016036379614428363175530, 6.164551965375185009492886191801, 7.26031331494109053900942649556, 8.8316185501020430335474020310, 9.66789601255563929905178474588, 10.488217055192857496061114610563, 12.06250972048453940767970782816, 12.90167010255530315271051811419, 14.09973243235085732245841952726, 15.867873882133858468792498613151, 16.88454453878046939372622884008, 17.47653713405968746962195059033, 18.75823463919911895627972606067, 19.83563436905113790217531732046, 20.5595639146180951132667355162, 21.70603534211830062800304927253, 22.616789226696882167378242390468, 24.23273444136979661388249912638, 25.43817675804422364284227190676, 25.848890002404748013082084707786, 27.11429483010897506988684530350, 28.19508874794290270118634005518, 28.89386233425548417165021032748

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