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Properties

Label 1-3528-3528.277-r0-0-0
Degree $1$
Conductor $3528$
Sign $0.708 - 0.705i$
Analytic cond. $16.3839$
Root an. cond. $16.3839$
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  + (−0.826 − 0.563i)5-s + (0.988 − 0.149i)11-s + (0.988 − 0.149i)13-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (0.733 − 0.680i)29-s + 31-s + (−0.955 − 0.294i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (0.623 + 0.781i)47-s + (−0.955 + 0.294i)53-s + (−0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)5-s + (0.988 − 0.149i)11-s + (0.988 − 0.149i)13-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (0.733 − 0.680i)29-s + 31-s + (−0.955 − 0.294i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (0.623 + 0.781i)47-s + (−0.955 + 0.294i)53-s + (−0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(16.3839\)
Root analytic conductor: \(16.3839\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3528,\ (0:\ ),\ 0.708 - 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.573383407 - 0.6500759102i\)
\(L(\frac12)\) \(\approx\) \(1.573383407 - 0.6500759102i\)
\(L(1)\) \(\approx\) \(1.070079071 - 0.1743399000i\)
\(L(1)\) \(\approx\) \(1.070079071 - 0.1743399000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.826 - 0.563i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.955 - 0.294i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (-0.0747 - 0.997i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.955 + 0.294i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + T \)
83 \( 1 + (0.988 + 0.149i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.98541696456678008453183849634, −18.09738800066647225629914666330, −17.62775035629658973370818820604, −16.58077429482247886581465862228, −16.0609583648852235472828032948, −15.404913600741506067614975328733, −14.70427954785137274653380103226, −14.04128595620551694011439106159, −13.41578083795184683930428633981, −12.41352264032805960736825633376, −11.7324737967119339951020216557, −11.29106307491683151684216340024, −10.55163086980442121052629163796, −9.69871062935087619279585707659, −8.8721859169009726796348159729, −8.274508642669676042044733449772, −7.414504479214910710696566744537, −6.658070330107321383842558916480, −6.26157994475135726317865267733, −5.00153098488460346679533040691, −4.33771520553605874772598533052, −3.44755949123262376825751636975, −3.01462600613276043009785515179, −1.72805135350625744479853746636, −0.881167292662699898235126426533, 0.70598516529528186717492779944, 1.34819127046295174183173403166, 2.57853308373191995889078351612, 3.54211051342889757932249768917, 4.14641236165543411566052924740, 4.81894468878130953700332772690, 5.78462433282443384742116234576, 6.61392109079264243528253260111, 7.22822178330872645976417057464, 8.22339012211895984184283448325, 8.82522312058415226370960865309, 9.1764301986679098674086737805, 10.44575605255029916974780620060, 11.017023629586874200361503471435, 11.77902560134063731628321578209, 12.24298167609267195355207447335, 13.2071873919310420062084801599, 13.652107895698528599782779147243, 14.57754232772912580490213514351, 15.509135731064580213826581383201, 15.68573605836553994290148654510, 16.56024833012894182331776830124, 17.38093216651562926351326895935, 17.69642267323874236061557023117, 19.04950174051389936053383123426

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