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Properties

Label 1-55-55.39-r1-0-0
Degree $1$
Conductor $55$
Sign $-0.944 - 0.329i$
Analytic cond. $5.91057$
Root an. cond. $5.91057$
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  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.309 − 0.951i)18-s + (−0.309 − 0.951i)19-s − 21-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.309 − 0.951i)18-s + (−0.309 − 0.951i)19-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(5.91057\)
Root analytic conductor: \(5.91057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (1:\ ),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05981301792 - 0.3525190259i\)
\(L(\frac12)\) \(\approx\) \(0.05981301792 - 0.3525190259i\)
\(L(1)\) \(\approx\) \(0.5014228556 - 0.1508186928i\)
\(L(1)\) \(\approx\) \(0.5014228556 - 0.1508186928i\)

Euler product

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

−33.69997403418889232107989557795, −32.08510006797704475383626170833, −31.13037911937757639539723709407, −29.67709788796490188511415100858, −28.54255360439969281661726113332, −27.77643524115132409303466794251, −26.87992219324888793062847784134, −25.775543125069012847037900930293, −24.52878977720746581892832159755, −22.50698229548014826685594160599, −21.716046743116397345169509750919, −20.715816848598324875805906804477, −19.546519892866705261509534114137, −18.11260388230741889924484825876, −17.154344984130186142658401952205, −15.90478078375845393134749594289, −14.81231768675594721192763790330, −12.57593411523242373229252765049, −11.510393861211950373164277075974, −10.32469146378187742702801373554, −9.23720757138289239721671279294, −8.03262163961509264982220075442, −5.870467103222237885267492440091, −4.08808499535580342142771702192, −2.368397655073377676384851017839, 0.25396735726581319583763022293, 1.95779564955139357433664729417, 4.98365368463321411777403296263, 6.721456329287944239799192059201, 7.42318772367692149892888808946, 8.863805784562759565632085879959, 10.53274542677448301560559419405, 11.68850168008681784339069638468, 13.49041275534962744782500668876, 14.5247313421230238130241873748, 16.267148614110196541867166392668, 17.31068144158782957956196627750, 18.095654966785499951832374649333, 19.42805187470451901379473121107, 20.215473546798544778079782840186, 22.3599034477932029067739251241, 23.85588289402763007188709470733, 24.13681837937353467684243184485, 25.5625295827049554429195913277, 26.59755826947429402793273852978, 27.77691363115174992141634399061, 29.03789338985930729590399148424, 29.74305878514524301222770262357, 31.13049374201423307289014179609, 32.68810094615847792152106184955

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