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Properties

Label 1-104-104.69-r0-0-0
Degree $1$
Conductor $104$
Sign $0.859 + 0.511i$
Analytic cond. $0.482973$
Root an. cond. $0.482973$
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)3-s + 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + 21-s + (−0.5 − 0.866i)23-s + 25-s − 27-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + 21-s + (−0.5 − 0.866i)23-s + 25-s − 27-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.482973\)
Root analytic conductor: \(0.482973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (0:\ ),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288691360 + 0.3541753007i\)
\(L(\frac12)\) \(\approx\) \(1.288691360 + 0.3541753007i\)
\(L(1)\) \(\approx\) \(1.291249177 + 0.2501954824i\)
\(L(1)\) \(\approx\) \(1.291249177 + 0.2501954824i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−29.660863332731324942413666526689, −28.82687424016998694947203978869, −27.827185010921866608608998780351, −26.183952813776923229144697795545, −25.454370770400769994206342534105, −24.68692144833118609006413399824, −23.730027452685174125172927551574, −22.35623470312590826791790717213, −21.20006426423752059882264236121, −20.34099107269857651734273265653, −19.03835222774413081820218015181, −17.8861339601659023626371962781, −17.607783996096574856136222338957, −15.61529440100672636384711811087, −14.56031988931439039735250853248, −13.511268100249686205616823139608, −12.62200893120899447147747676213, −11.40904551344656881498173382550, −9.69639223867933151650055163623, −8.77539317905238582073472233273, −7.4542251285324526000363880529, −6.22152181351879384100683978153, −4.96450678752792912527998585715, −2.66582896025773568177154364920, −1.83671570534581056424627007949, 1.97907815088591352776868015566, 3.56738605753523571469733953735, 4.87694529924387650397855641807, 6.16580115473281837296560030197, 7.97922932543144328821640871969, 8.996625810336510938766628070987, 10.4219472408708471009070416485, 10.79139040032724931020498032332, 12.91673434139402087338801804816, 14.018985809867092670334959000001, 14.64128957029327589816662713904, 16.207706636072382336397519448637, 16.971096852278158221962301372017, 18.17105865990686052049339728585, 19.605166085965235388409903353459, 20.70738441400063213283540007900, 21.35928941018932146496213375818, 22.281344209761556947616247352111, 23.7253388291041530634555556476, 24.831005014865795851054914724030, 25.99287881506459123733209241462, 26.605641534689777448654442904577, 27.63918545054613781160558403847, 28.84933101595610237926930517079, 29.81586820532432805112310685953

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