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Properties

Label 1-621-621.155-r0-0-0
Degree $1$
Conductor $621$
Sign $0.0238 + 0.999i$
Analytic cond. $2.88391$
Root an. cond. $2.88391$
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  + (0.987 + 0.158i)2-s + (0.950 + 0.312i)4-s + (−0.0792 + 0.996i)5-s + (−0.110 + 0.993i)7-s + (0.888 + 0.458i)8-s + (−0.235 + 0.971i)10-s + (0.873 − 0.486i)11-s + (−0.916 − 0.400i)13-s + (−0.266 + 0.963i)14-s + (0.805 + 0.592i)16-s + (0.928 + 0.371i)17-s + (−0.928 + 0.371i)19-s + (−0.386 + 0.922i)20-s + (0.939 − 0.342i)22-s + (−0.987 − 0.158i)25-s + (−0.841 − 0.540i)26-s + ⋯
L(s)  = 1  + (0.987 + 0.158i)2-s + (0.950 + 0.312i)4-s + (−0.0792 + 0.996i)5-s + (−0.110 + 0.993i)7-s + (0.888 + 0.458i)8-s + (−0.235 + 0.971i)10-s + (0.873 − 0.486i)11-s + (−0.916 − 0.400i)13-s + (−0.266 + 0.963i)14-s + (0.805 + 0.592i)16-s + (0.928 + 0.371i)17-s + (−0.928 + 0.371i)19-s + (−0.386 + 0.922i)20-s + (0.939 − 0.342i)22-s + (−0.987 − 0.158i)25-s + (−0.841 − 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0238 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0238 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.0238 + 0.999i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.0238 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872451471 + 1.828298147i\)
\(L(\frac12)\) \(\approx\) \(1.872451471 + 1.828298147i\)
\(L(1)\) \(\approx\) \(1.734614882 + 0.7849828478i\)
\(L(1)\) \(\approx\) \(1.734614882 + 0.7849828478i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.987 + 0.158i)T \)
5 \( 1 + (-0.0792 + 0.996i)T \)
7 \( 1 + (-0.110 + 0.993i)T \)
11 \( 1 + (0.873 - 0.486i)T \)
13 \( 1 + (-0.916 - 0.400i)T \)
17 \( 1 + (0.928 + 0.371i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.745 + 0.666i)T \)
31 \( 1 + (-0.386 - 0.922i)T \)
37 \( 1 + (0.995 + 0.0950i)T \)
41 \( 1 + (-0.902 + 0.429i)T \)
43 \( 1 + (0.605 + 0.795i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (-0.805 + 0.592i)T \)
61 \( 1 + (-0.678 - 0.734i)T \)
67 \( 1 + (0.0158 + 0.999i)T \)
71 \( 1 + (-0.981 + 0.189i)T \)
73 \( 1 + (0.928 - 0.371i)T \)
79 \( 1 + (0.553 - 0.832i)T \)
83 \( 1 + (0.902 + 0.429i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.701 - 0.712i)T \)
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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

−22.93323393786826125291606182471, −21.94497112063562508470585002745, −21.15420174802889633883348442106, −20.39056920324136297520227837795, −19.72480749414690325543107069067, −19.18086628799632794222239198203, −17.39090769854961198078619379280, −16.87227797858793383539207469991, −16.17775096508855259771123969849, −15.11397431555725334837664749137, −14.22476310101619690985786244978, −13.62807059802476807203570423088, −12.48470997043597043972694470714, −12.20019540051500950662861331823, −11.097766971568928366303796223820, −10.05292294400973582273510873746, −9.279986344687765293434201184267, −7.86228352580719047066988211595, −7.058144983893973226421453663724, −6.1397297897202532914373998210, −4.82112439495114092256122789356, −4.43697171266923236136005854290, −3.42826093256832903140055532160, −2.02450377374184975717246165268, −0.972804694348149600961899083732, 1.85205262571263397085755790154, 2.85041802324520829118036386004, 3.545855410986740582251870109277, 4.74024096535592817198267459105, 5.93790207199645313218644493179, 6.337221992928922343595561139691, 7.46676051466711649382766129470, 8.33723700504616877360779569475, 9.700540450215673990974536826285, 10.654154410275128945772538600221, 11.60321156599019878257289358349, 12.21331274158121854388474308388, 13.069159996126791273373939429, 14.32979594257327580075651740068, 14.71033387817756611964739018393, 15.32933474754537959634398175174, 16.438540380756975915685566903580, 17.18855615949688393176995391653, 18.37664980041662297474368013175, 19.2323169558455770041117983189, 19.81909391477050291343972981092, 21.12932310924255775333316276125, 21.87410889038949049426510340769, 22.17944684884750867019095136115, 23.10582818439824505587442932555

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