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Properties

Label 1-108-108.43-r1-0-0
Degree $1$
Conductor $108$
Sign $0.549 + 0.835i$
Analytic cond. $11.6062$
Root an. cond. $11.6062$
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  + (0.766 + 0.642i)5-s + (0.939 + 0.342i)7-s + (−0.766 + 0.642i)11-s + (0.173 − 0.984i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (0.173 + 0.984i)29-s + (0.939 − 0.342i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)5-s + (0.939 + 0.342i)7-s + (−0.766 + 0.642i)11-s + (0.173 − 0.984i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (0.173 + 0.984i)29-s + (0.939 − 0.342i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.939 + 0.342i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(11.6062\)
Root analytic conductor: \(11.6062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 108,\ (1:\ ),\ 0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.810001015 + 0.9759438689i\)
\(L(\frac12)\) \(\approx\) \(1.810001015 + 0.9759438689i\)
\(L(1)\) \(\approx\) \(1.298160471 + 0.3136786286i\)
\(L(1)\) \(\approx\) \(1.298160471 + 0.3136786286i\)

Euler product

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

−29.00435193219455885448652755190, −28.38467833269212171974241298763, −26.99775354408657318239616068339, −26.23824163008815285246529471101, −24.81009361805943806224536939808, −24.21978985679445417711051953006, −23.14656868116882023052118740983, −21.499931587456943329946946084197, −21.1037822851129965696335938637, −19.969906271422441723051772901075, −18.5267624862461912589901226474, −17.58295511108621172291901540487, −16.629771955151695431966365726290, −15.478393590388613636210297386, −13.85180087135884736079346607858, −13.493169188831011028472887366606, −11.82567690715714851233871592143, −10.81471151859591549877085063219, −9.3985890668342377464178080872, −8.41845857200626963302710350642, −6.99671220877195847129046409559, −5.439626466437897763888298489333, −4.51468893419212955130435305590, −2.48139098358185944900494646345, −0.96644246025284188955484764055, 1.661561201945086902433623029554, 2.97158713115382502711298142916, 4.89361724941464899797393200572, 5.963474932834986870482347591503, 7.43320583619083792277223822256, 8.60212679155053824213066792331, 10.15945705693242762646816780263, 10.86563285449258858649289157472, 12.37858315635053232343586156327, 13.51869680550996941304744032522, 14.726311556743277246002798952486, 15.434028557325221534546624870278, 17.18626259920945868343862526386, 17.95610895671033503653937119126, 18.75134389934927997737396587756, 20.398326912419582539570699287555, 21.154772108130171472825677818606, 22.23309680871635446110224272803, 23.21315220284428847860920705214, 24.522462617602207113986806101793, 25.37586675076306898369852427843, 26.36474475848236952702452974896, 27.443342969260815384905028843237, 28.51174507611553906604977008593, 29.49566437509931049651543570465

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