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Properties

Label 1-55-55.4-r0-0-0
Degree $1$
Conductor $55$
Sign $0.822 - 0.568i$
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  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s − 12-s + (−0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + 21-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s − 12-s + (−0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8874743712 - 0.2767391312i\)
\(L(\frac12)\) \(\approx\) \(0.8874743712 - 0.2767391312i\)
\(L(1)\) \(\approx\) \(1.011577065 - 0.2605437617i\)
\(L(1)\) \(\approx\) \(1.011577065 - 0.2605437617i\)

Euler product

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

−33.49948260830639171807634334188, −31.822265518769710825734035303849, −31.46852442707584676679714354135, −30.07806436953977988913687765728, −28.55214878570009260981190798841, −27.25716882727417861287977812895, −26.28831306322172644779501255201, −25.15794649645322535915054069286, −24.397791196807406163186359610, −23.48762508913542538158525165617, −21.8228304933916129897753590640, −20.36299935200787931132788284315, −18.892029377946614583073811785488, −18.26731912462664013309545939453, −16.90877920733558120179152523960, −15.34875904381573049807370530811, −14.46384017437891978261890577709, −13.45661926681055242809637216702, −11.8118252658395820773634604224, −9.65923360035358874998508347667, −8.55445084281779651911283230611, −7.5457066593740960106807721353, −6.1739096343193584086025142671, −4.39560333377502517978436128830, −1.971448959592245139543691351071, 1.99667011438049429979367967689, 3.61939879193141055867415378074, 4.82770641940531080252354209309, 7.756868236998142656203630729035, 8.706775998583808217769480336450, 10.189293925504757294865244026834, 10.96742498374779404832200407491, 12.74727808666989897976320159439, 13.93976664877493562105266950396, 15.10686625051322361705336870071, 16.87329332309144695559088663513, 18.00536664344550772669720782242, 19.560915611623006716801028801319, 20.22773333891576712674693451361, 21.29612351392970691878104624523, 22.21038148346257492280334461505, 23.83410430655072663162705406317, 25.47123473733393843780748900906, 26.54503379534831116338914749913, 27.39209869300254187503985681420, 28.28061126150332577244767077548, 30.0308861233454996119172354424, 30.50156761839451478970885696910, 31.81034538560283580572690419, 32.66927930203567886792795546578

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