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Properties

Label 1-208-208.155-r1-0-0
Degree $1$
Conductor $208$
Sign $0.923 - 0.382i$
Analytic cond. $22.3527$
Root an. cond. $22.3527$
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  + i·3-s − i·5-s − 7-s − 9-s + i·11-s + 15-s + 17-s − i·19-s − i·21-s + 23-s − 25-s − i·27-s − i·29-s + 31-s − 33-s + ⋯
L(s)  = 1  + i·3-s − i·5-s − 7-s − 9-s + i·11-s + 15-s + 17-s − i·19-s − i·21-s + 23-s − 25-s − i·27-s − i·29-s + 31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(22.3527\)
Root analytic conductor: \(22.3527\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 208,\ (1:\ ),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.374057656 - 0.2733170613i\)
\(L(\frac12)\) \(\approx\) \(1.374057656 - 0.2733170613i\)
\(L(1)\) \(\approx\) \(0.9717157278 + 0.06905771053i\)
\(L(1)\) \(\approx\) \(0.9717157278 + 0.06905771053i\)

Euler product

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

−26.37816260239144362082765630215, −25.56480631940391830102507660527, −24.85431070974044429356652349875, −23.58528697203971494143465183892, −22.924915908626228263229025868729, −22.12740065555552451680212059501, −20.90441433247561576347240680128, −19.44076798766966341614575396747, −18.99997035231843394576508455731, −18.312753037057862485649575336846, −17.0518830967758721725532049246, −16.13700283121377299863927812449, −14.72530008061590645220288879065, −13.94234163388637813866722355556, −12.99497674719245871993803300400, −11.987611215778755545251067597602, −10.9428305624441614041973569648, −9.85758450783660796821525862361, −8.46773593967013029605452108819, −7.379690175688721463966245705017, −6.44255287517186322590029616367, −5.696997825442674445409644126544, −3.430123025050757914874379290005, −2.77185761703274512904230036956, −1.03165662653792363061492328525, 0.58870096276584900675119084208, 2.63392845030781682133015086032, 3.97652587864385600238488403413, 4.907701469115428721219783159231, 5.97096938226038699714235482321, 7.49123697760814830152609685458, 8.946204135940911181575206862876, 9.52641332310625782521286746289, 10.45616356516396188059203089150, 11.875750950327308930814678222550, 12.74044031127364466713932537207, 13.83315148199233780031776141003, 15.23119293758720806843717484847, 15.8164830134795879423727460016, 16.83763598212505969442545402890, 17.438426747935972873359499673699, 19.147739394902322588403083876952, 19.98633970369618739572159751738, 20.8087272127691964938213366483, 21.596617415392594146928951080130, 22.79667815350633206280749063687, 23.31288241817283964781619122256, 24.8139073422945230357895102850, 25.58949015434282815165723509077, 26.40632671358966273339611731106

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