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Properties

Label 1-55-55.8-r0-0-0
Degree $1$
Conductor $55$
Sign $0.985 - 0.169i$
Analytic cond. $0.255418$
Root an. cond. $0.255418$
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  + (0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.309 − 0.951i)6-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s − i·12-s + (−0.951 − 0.309i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + (−0.809 + 0.587i)19-s − 21-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.309 − 0.951i)6-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s − i·12-s + (−0.951 − 0.309i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s + (−0.809 + 0.587i)19-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.247352780 - 0.1062066600i\)
\(L(\frac12)\) \(\approx\) \(1.247352780 - 0.1062066600i\)
\(L(1)\) \(\approx\) \(1.384027223 - 0.06411751476i\)
\(L(1)\) \(\approx\) \(1.384027223 - 0.06411751476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−33.11474880017254542783164963920, −31.97423204837713617473263986924, −31.19435442750361610112262516913, −29.78443828353706864957922104306, −28.65666410982032812564902096240, −27.87899466598695726792410582451, −26.51482659267726019914667266942, −24.832174890122814435788114485470, −23.87696378801827405158563867195, −22.54963864488306443129317487711, −21.76793831691038202869501814261, −20.93005829832175522925431843971, −19.60606160295174319655537087408, −17.927605482249229570164755834998, −16.4499819458379640027018815239, −15.25453856577886381565558045646, −14.47642066511400246301301528974, −12.64207897334552310793388840469, −11.59507781308331844231805470245, −10.59836497973071322411584132896, −9.059823396217572868533787355645, −6.70818183117028235697318067231, −5.281864042887035461200680907486, −4.36456043783324990822419228310, −2.46838343379528889907541660924, 2.07366721334378857674144358529, 4.26981074670714828746090483678, 5.6404744869139935069803192426, 7.01980041451009374580179115890, 7.96719660406043990749351994067, 10.65802019420099425487110199039, 11.74832315592209633624343508384, 12.954730059179764571783288361879, 13.91479321622556160090030789300, 15.22862535290310598521983933982, 16.92456298348977877228650014870, 17.47410042040384150988551225586, 19.32769577986783592592925561924, 20.5389857939140680017374093762, 21.96618227848747642377234965571, 22.96617571104121034521339677970, 23.98346832303542200586630662808, 24.63298958390636410199750162619, 26.002520115029621886196469517330, 27.51338259755597056317384996786, 29.15284212087660361390661699101, 29.83804565163015442224492475876, 30.773048063325668023510806408624, 31.890315105698219145555395506386, 33.369691919099007219888527893468

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