Properties

Label 1-104-104.101-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: $\chi_{104} (101, \cdot )$
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

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

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