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Properties

Label 1-55-55.28-r0-0-0
Degree $1$
Conductor $55$
Sign $0.976 - 0.215i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ 0.976 - 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5503282578 - 0.06006559246i\)
\(L(\frac12)\) \(\approx\) \(0.5503282578 - 0.06006559246i\)
\(L(1)\) \(\approx\) \(0.6508393320 - 0.09579938490i\)
\(L(1)\) \(\approx\) \(0.6508393320 - 0.09579938490i\)

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

−33.30613409542745829584308529346, −32.657635888987720611772534603338, −30.78293586622161980847311911549, −29.6696737130935714119221168650, −28.22496393647885817073566762062, −27.704698222162852117745505886018, −26.47157905840968837762246234263, −25.06262273896952541362880866865, −24.012290634046437396228216666536, −23.31550644798732076554593923303, −22.00943660701152490758765551264, −20.29739022898702522895838471467, −18.72745929299690548947901030783, −17.81013543486729266904562299457, −16.99813772665704048460146264250, −15.78745689388763468654311484011, −14.46207866133616822788094775616, −12.980721655873408015297192238706, −11.203122543034017308760966191823, −10.316405811168342519409553276597, −8.41608252142690754550326423991, −7.262392577107812315261258607162, −5.894312604463391549488023422141, −4.721092282288538746306197032125, −1.26959791632776242148227469179, 1.576080194815700384002094861592, 3.87749190417858927367172114376, 5.374053218279447064050464018109, 7.37511120664568264868424324156, 8.97505960390645510222355316244, 10.296984907341494325431348299019, 11.48606060504573254322808598522, 12.137346284523891890557672614056, 13.91980790937408921773074116794, 15.83829864921251847921921212137, 16.98333566315694287664696984133, 18.02966278089450555364328300559, 18.886465466451958601455326766998, 20.73939072726247016378280882456, 21.327354863213187222358069403, 22.55064019834404266787481752143, 23.75454133536486658095223032828, 25.321038915294105907814249326214, 26.88885155734373490287118306015, 27.552574378625779646171785447442, 28.54288850840372639791911574897, 29.46320726209551709904727506834, 30.60489507704438487106455275473, 31.71800091889851299853320338655, 33.468039070610996698856180472021

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