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Properties

Label 1-108-108.23-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: $\chi_{108} (23, \cdot )$
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

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

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