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Properties

Label 1-117-117.106-r1-0-0
Degree $1$
Conductor $117$
Sign $0.948 + 0.317i$
Analytic cond. $12.5733$
Root an. cond. $12.5733$
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  − i·2-s − 4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 − 0.866i)10-s + i·11-s + (−0.5 + 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + 22-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 − 0.866i)10-s + i·11-s + (−0.5 + 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + 22-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.948 + 0.317i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ 0.948 + 0.317i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339265686 + 0.2183812651i\)
\(L(\frac12)\) \(\approx\) \(1.339265686 + 0.2183812651i\)
\(L(1)\) \(\approx\) \(0.9859949366 - 0.2002026546i\)
\(L(1)\) \(\approx\) \(0.9859949366 - 0.2002026546i\)

Euler product

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

−28.80586477327849119826238770364, −27.81398644452305627273631839926, −26.628511544198338392781413702943, −25.6394296602350720931419177107, −24.95421293489689064789405639216, −24.09194683128895857470212972780, −22.83669049161270270588150714099, −21.91973705621557231594931525200, −21.001695578368051130819073160848, −19.27169653496569329746672643272, −18.4431820947190445014852267666, −17.193631911318029601159201288971, −16.4249487056856933313457526948, −15.50093748157860656350491390361, −14.095971191289780714665427644797, −13.32095321422498616788501231584, −12.28950251398656056454436195757, −10.29136143415388980069797477345, −9.18855570115095132602417900239, −8.43528942087232810895684361243, −6.686719374651559078418264800688, −5.89324714182020443950237771621, −4.74761611135143393880407700383, −2.923934607705529207115175825796, −0.60077842141066253819423415603, 1.495222472975189758031998897512, 2.839042027492872718889465237074, 4.11305005245634305511539999084, 5.70168492896674992827679377591, 7.07143509572273778669755118196, 8.82497133219722380107851450621, 10.16594348176202610820296859463, 10.35463610408408146686002004028, 12.11316039262986405909506856661, 13.05366803430142730989608366514, 13.96585810925824154164918220201, 15.1180126643350949141932498847, 16.977228590665383239125908762574, 17.66137781044122961575990756096, 18.910895671259831438653525514423, 19.68646671890614874024428920958, 20.908069417343473157231682844417, 21.690419335794340929687350471161, 22.805613132596858960289792165663, 23.39376857154992221907288350162, 25.36400719878855120711888984530, 25.98509756257485908888202572530, 27.08468617964026313755438764702, 28.293908274614124662789129997987, 29.09360612430937363858656575622

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