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Properties

Label 1-55-55.13-r0-0-0
Degree $1$
Conductor $55$
Sign $0.629 - 0.776i$
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  + (0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.809 − 0.587i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s − i·12-s + (−0.587 + 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (0.309 − 0.951i)19-s − 21-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.809 − 0.587i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s − i·12-s + (−0.587 + 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (0.309 − 0.951i)19-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ 0.629 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.192152778 - 0.5683360877i\)
\(L(\frac12)\) \(\approx\) \(1.192152778 - 0.5683360877i\)
\(L(1)\) \(\approx\) \(1.367161832 - 0.4921559486i\)
\(L(1)\) \(\approx\) \(1.367161832 - 0.4921559486i\)

Euler product

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

−32.81775300270501478263209160742, −32.35988449182578372633574165422, −31.23170598093495962523255814615, −30.27388592309963862736214766931, −29.17058353195670691464270385920, −27.07026665686668180007731211166, −26.23205727777114118578723317800, −25.24414933925523630238077066694, −24.40661648825982132527334037762, −23.117226861563609854925235841910, −22.028021260785878877016376047066, −20.62682490981437618920817261271, −19.49901293657540727907530684135, −18.0772383544657719201455041691, −16.63278033206425081909111681395, −15.41069253895666143732195729084, −14.412603912258286332720722600843, −13.18248344968332986421255827115, −12.44389357195310560611012782047, −9.97977997982853140298458611693, −8.53151867368148336758311008261, −7.37032961757289817979089427493, −6.133185578768820483466016677923, −4.1181216631115161517642724414, −2.83895213214041965151291955511, 2.28731240708878105101870365551, 3.48049497381763519703576741018, 4.94757883119288757050877781449, 6.907478779328130428659463815093, 9.10653097517641382741848624347, 9.75631827997916634653930441608, 11.40146632603446990577870459196, 12.889259892724776663602661339346, 13.79954790124082075147025725181, 15.02996053273831534557209392941, 16.11923173960379957553259457571, 18.35828167907059749058880077129, 19.50148641711639883616358588448, 20.1324415196125055660232690639, 21.58200868631092639303671544830, 22.19919927751599716355206426409, 23.76216384729566426897249906768, 24.9840065246505568100378826235, 26.255374292473091789319693071248, 27.390815113210587343147436031345, 28.7189161275482176949370759692, 29.668427310830459898208811655721, 31.0237486662077113728779319868, 31.67008515347460914163819630178, 32.581577953958758744152623949885

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