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Properties

Label 1-208-208.11-r0-0-0
Degree $1$
Conductor $208$
Sign $0.985 + 0.168i$
Analytic cond. $0.965947$
Root an. cond. $0.965947$
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  + (−0.866 − 0.5i)3-s − 5-s + (−0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + 21-s + (0.5 − 0.866i)23-s + 25-s − i·27-s + (0.866 + 0.5i)29-s − i·31-s + (−0.866 + 0.5i)33-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s − 5-s + (−0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + 21-s + (0.5 − 0.866i)23-s + 25-s − i·27-s + (0.866 + 0.5i)29-s − i·31-s + (−0.866 + 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(0.965947\)
Root analytic conductor: \(0.965947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 208,\ (0:\ ),\ 0.985 + 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6486111181 + 0.05510617148i\)
\(L(\frac12)\) \(\approx\) \(0.6486111181 + 0.05510617148i\)
\(L(1)\) \(\approx\) \(0.6804045889 + 0.02567679178i\)
\(L(1)\) \(\approx\) \(0.6804045889 + 0.02567679178i\)

Euler product

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

−26.895070031859523773235015150131, −26.01416603599907603266633719681, −24.76244633750001616140104118456, −23.47317309855267714842733211233, −22.996249754367830309475185368133, −22.34349837882590155185695323888, −21.096211954457231762064634236291, −20.00118236094038170109769785930, −19.2981304416451546522428180927, −17.960830574395642582521619306343, −17.07990191883741527496472294015, −15.96746374637512129273823951004, −15.62602926522967650704035327685, −14.27632123810259227792943637747, −12.79679090939915412612519050543, −11.97676655548919817817756769542, −11.122080238955543499837478778246, −9.985708401647182830611515864736, −9.14100301060916510476876459198, −7.34035014866941576920742563088, −6.77241914632092562260533893812, −5.21817500669940254288093482805, −4.202247015217806994630059375867, −3.21073964180408264471864256598, −0.761818647327825859558477928494, 1.01445438607741153522992768164, 2.989632014550543550405733539328, 4.22658173468742786639098161142, 5.72574874501688766732765206323, 6.501520735547277186464617899, 7.69689340906276305957903326156, 8.73768689481711661167154291380, 10.24210538865563060891202754149, 11.2650554611308493388294447063, 12.20401016577454374374444534098, 12.76169462568217996497813752104, 14.157104163615534720528184872277, 15.47239810612119998226525976891, 16.36621831202316843983810023367, 16.96627532394883169009319795817, 18.534254377892743363631810985630, 18.98095528304749128452693172355, 19.8130764879516531418456259078, 21.317379514885140256700382977602, 22.43552410116455117934743816882, 22.86614871872993349017056569494, 23.97868161759143537142252397037, 24.61860515825008470654149405700, 25.78062830742652744022307979945, 27.0208875722977630377589486181

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