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Properties

Label 1-117-117.22-r0-0-0
Degree $1$
Conductor $117$
Sign $0.815 - 0.578i$
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  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + 11-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + 22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + 11-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + 22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647660323 - 0.5253185935i\)
\(L(\frac12)\) \(\approx\) \(1.647660323 - 0.5253185935i\)
\(L(1)\) \(\approx\) \(1.626611814 - 0.3048064331i\)
\(L(1)\) \(\approx\) \(1.626611814 - 0.3048064331i\)

Euler product

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

−29.55973354326705430174512278275, −28.51018826195320559399662234292, −27.30928950668749952766659173202, −26.00284321324701401005907650395, −25.13497411661965659976247721233, −24.15074763951952062980694544373, −22.9092487114442818774631565547, −22.28058137833218788849357382325, −21.53761144472901597100569564441, −19.9904666093900707562750734914, −19.2919259018650777126081182483, −18.0385942196396164184941951042, −16.40521918882784541990604923330, −15.46230637307917012974217177478, −14.66534880335549437098627710877, −13.601865444899091371439651541519, −12.2041870839215931604851324676, −11.56070342216743678776165403466, −10.30852764955445230256492723007, −8.68277230886967752193625208309, −6.95494350551748548554991332071, −6.355082216406049085171201139457, −4.75914659887944455455381294469, −3.41339982727858896871146016116, −2.365101615483467487660556627835, 1.550931693334795453524122655548, 3.68876591240343172823050680437, 4.28590970271190754365622873290, 5.855970794160857870975365405645, 7.02734097241849911210185459537, 8.31379625477152107681806152691, 9.93786101468673537721310290518, 11.27788147775694386928862756190, 12.33585001552931758315713296040, 13.19179589135443067785005374704, 14.26489918132470015313309601733, 15.46553584790734188436127399810, 16.51180737929382182098113005025, 17.19465014536954908951238315766, 19.44287928572404724277406962631, 19.874037599513999628532804262852, 20.93565247999767526204804538285, 22.06550050186118942205651947912, 23.142944628049649465826873479272, 23.826945944068802240878935427574, 24.81183118270475863204545684974, 25.80326797115326375886656138987, 27.165512812999879065377936898219, 28.275740804171368500687118574161, 29.33883471426740245140215257500

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