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Properties

Label 1-108-108.47-r0-0-0
Degree $1$
Conductor $108$
Sign $0.835 - 0.549i$
Analytic cond. $0.501549$
Root an. cond. $0.501549$
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  + (0.939 − 0.342i)5-s + (−0.173 − 0.984i)7-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.766 + 0.642i)29-s + (−0.173 + 0.984i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.766 − 0.642i)41-s + (0.939 + 0.342i)43-s + (0.173 + 0.984i)47-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)5-s + (−0.173 − 0.984i)7-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.766 + 0.642i)29-s + (−0.173 + 0.984i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.766 − 0.642i)41-s + (0.939 + 0.342i)43-s + (0.173 + 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(0.501549\)
Root analytic conductor: \(0.501549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 108,\ (0:\ ),\ 0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094853404 - 0.3277775921i\)
\(L(\frac12)\) \(\approx\) \(1.094853404 - 0.3277775921i\)
\(L(1)\) \(\approx\) \(1.126831765 - 0.1842374946i\)
\(L(1)\) \(\approx\) \(1.126831765 - 0.1842374946i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−29.67707934994914119943882443926, −28.50964282763389929865459519048, −27.985444831417855735990345417038, −26.26096442562713301663726183477, −25.67498473249285138971951078041, −24.74741518683178788718419862787, −23.46094809347194360609576481617, −22.29825965243369902502319571540, −21.45200469974565297009578971627, −20.55310381247847855950711123697, −18.99948109013775543399662793621, −18.16545630410533212767832544680, −17.29676763366970155663897045081, −15.732623457976346920793229585867, −14.98049949962077010067574907650, −13.50503773267810861380410086716, −12.755333125265460828863473041495, −11.23262144730942396469896975297, −10.078473659752856022801092104488, −9.04192196446980154951856052532, −7.651651482912456007060071495022, −6.04572518440902329318357516223, −5.32327860782301098394204802442, −3.21589648178561447427254148499, −1.97397556744650814622441597697, 1.371312419122691024031208556861, 3.16092901682260669369830400347, 4.787839136986789858574279807314, 6.028942944455808057651984472685, 7.34363304737300193764380745022, 8.76250424408381766000382314874, 9.99786602360163773772391668261, 10.87751343680304406520293853723, 12.50835524306574796995318774403, 13.621330340851534730144539383278, 14.22316002919279342968375937143, 16.117514465345919996206594591144, 16.68743451114332155069085880908, 18.00021714013479262477936948731, 18.87429966637913848650248985498, 20.638238068041197471082119473526, 20.795299546444288424892625365600, 22.27817242193329344255713283225, 23.37901025522930232798860963315, 24.31208706979624078067497235759, 25.532216148293269676451858995142, 26.28945287506056445993701973233, 27.37472538112387279542258645625, 28.86436449175072382158378968775, 29.20349273881966059122508538408

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