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Properties

Label 1-117-117.56-r1-0-0
Degree $1$
Conductor $117$
Sign $0.774 - 0.632i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ 0.774 - 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4355518306 - 0.1553725750i\)
\(L(\frac12)\) \(\approx\) \(0.4355518306 - 0.1553725750i\)
\(L(1)\) \(\approx\) \(0.5432567673 + 0.2056561630i\)
\(L(1)\) \(\approx\) \(0.5432567673 + 0.2056561630i\)

Euler product

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

−28.93371691285111233742486440797, −28.23630518262617099138482294616, −27.1857040195035998486035362604, −26.2795090728829703926502070744, −25.23272605425165442784454999606, −23.86995900455737512786726310471, −22.82495525686101056133802898364, −21.656414646882644436742789542869, −20.74912878504696068523277563554, −19.634968257818505698959380763908, −19.10214325217374933935976648549, −17.820902519172366433456605422769, −16.412126342241722071811138399445, −16.07786172640811964033771732604, −13.99090061143698771669061660495, −12.76330221405652465233661186729, −12.22152724892209695658887201562, −10.82219212802754099606985624673, −9.73848075343171647570213333123, −8.620434588690448458969484340355, −7.70054557223286329848163830851, −5.77837749570914410788170821988, −4.10679753872046233104511469398, −3.05832547397080670409094594280, −1.1496778737537610183888273418, 0.26587582946407106953206515, 2.703872423506612583730792660425, 4.407322043249547273938476883020, 6.00420061699712747645780850574, 7.060362333563226278906116190546, 7.87069487192498296243969769991, 9.544073119226552242345225440866, 10.20623211290706116386292437927, 11.68572677049091603544880151393, 13.24877608829196933439360153622, 14.36613729903780057299077063705, 15.5478064579883807643760110528, 16.02566651118088356039045228103, 17.49857157240740388879664817937, 18.44036645513387128045192069701, 19.2727117956198799152042877521, 20.27583383104556073692297838094, 22.20759723442108477921426395299, 22.86791156191203410893867251188, 23.69365438330106152057391039224, 25.02887766117936675951644383542, 26.02581715956670806807505351522, 26.45967615368750800130950305041, 27.706017355308077836437813556753, 28.5823596468159564834131971481

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