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Properties

Label 1-108-108.95-r0-0-0
Degree $1$
Conductor $108$
Sign $0.0581 - 0.998i$
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  + (−0.173 − 0.984i)5-s + (−0.766 − 0.642i)7-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (−0.766 + 0.642i)31-s + (−0.5 + 0.866i)35-s + (−0.5 − 0.866i)37-s + (0.939 + 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.766 + 0.642i)47-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + (−0.766 − 0.642i)7-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (−0.766 + 0.642i)31-s + (−0.5 + 0.866i)35-s + (−0.5 − 0.866i)37-s + (0.939 + 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.766 + 0.642i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.0581 - 0.998i$
Analytic conductor: \(0.501549\)
Root analytic conductor: \(0.501549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 108,\ (0:\ ),\ 0.0581 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6144165308 - 0.5796721002i\)
\(L(\frac12)\) \(\approx\) \(0.6144165308 - 0.5796721002i\)
\(L(1)\) \(\approx\) \(0.8485765085 - 0.3245992846i\)
\(L(1)\) \(\approx\) \(0.8485765085 - 0.3245992846i\)

Euler product

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

−29.65256498854089114210518185174, −29.02300079736038385607683762134, −27.66037905244055784074541013663, −26.79100792882036665530614332113, −25.62227984670828132539961204444, −25.01420198334561672380798712293, −23.40506984486889816438160837881, −22.55078829915945513685728898757, −21.84548750589014811281973531307, −20.40933704459019538260544965569, −19.20615447948160405733110941323, −18.52076051079686394822511952333, −17.309347949423411556307404891787, −15.95290271520698656152081528894, −14.98504829719536992475813025332, −14.05194592634709493336621497035, −12.48679266615840150047836627057, −11.67713808593340777883141404298, −10.12708522661485870992560812692, −9.37464280233212473772312566550, −7.54036985891803344985700986146, −6.69497141687295195217784445133, −5.23135287643856390755919429374, −3.4959945264182356740835830522, −2.307543097953968357765214575752, 0.86265631827596824109838692750, 3.08429665359403964217736272167, 4.468049020835512744481938917969, 5.802660529673255096840349047900, 7.26094382340804555741870667590, 8.54971075229711055829102305696, 9.6441961248293219076629207696, 10.90797796195453146719216500708, 12.363417042999333248437877363624, 13.12943638931542478043002888062, 14.34464349839000652715587860342, 15.85113749283445488369100986517, 16.64578219994070678814158664472, 17.52477720183252138253358226073, 19.313647574195472113815111418031, 19.74812633895018016899656151209, 21.01695068037910995774286068177, 22.069855352228012353552176878540, 23.28965532057962935120011576419, 24.21012743385602857922655305628, 25.09218990198217619660863539041, 26.41950946660571439407278317325, 27.21435020025855947262478044787, 28.45662619515928123176660166150, 29.22941020866405602077918933673

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