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Properties

Label 1-66e2-4356.515-r0-0-0
Degree $1$
Conductor $4356$
Sign $0.999 + 0.00144i$
Analytic cond. $20.2291$
Root an. cond. $20.2291$
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  + (−0.380 − 0.924i)5-s + (−0.640 − 0.768i)7-s + (−0.290 + 0.956i)13-s + (0.362 − 0.931i)17-s + (−0.774 + 0.633i)19-s + (0.928 + 0.371i)23-s + (−0.710 + 0.703i)25-s + (−0.964 + 0.263i)29-s + (0.905 + 0.424i)31-s + (−0.466 + 0.884i)35-s + (0.516 − 0.856i)37-s + (0.398 − 0.917i)41-s + (−0.235 − 0.971i)43-s + (−0.217 + 0.976i)47-s + (−0.179 + 0.983i)49-s + ⋯
L(s)  = 1  + (−0.380 − 0.924i)5-s + (−0.640 − 0.768i)7-s + (−0.290 + 0.956i)13-s + (0.362 − 0.931i)17-s + (−0.774 + 0.633i)19-s + (0.928 + 0.371i)23-s + (−0.710 + 0.703i)25-s + (−0.964 + 0.263i)29-s + (0.905 + 0.424i)31-s + (−0.466 + 0.884i)35-s + (0.516 − 0.856i)37-s + (0.398 − 0.917i)41-s + (−0.235 − 0.971i)43-s + (−0.217 + 0.976i)47-s + (−0.179 + 0.983i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00144i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ 0.999 + 0.00144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098550819 + 0.0007922864513i\)
\(L(\frac12)\) \(\approx\) \(1.098550819 + 0.0007922864513i\)
\(L(1)\) \(\approx\) \(0.8571432658 - 0.1402434193i\)
\(L(1)\) \(\approx\) \(0.8571432658 - 0.1402434193i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.380 - 0.924i)T \)
7 \( 1 + (-0.640 - 0.768i)T \)
13 \( 1 + (-0.290 + 0.956i)T \)
17 \( 1 + (0.362 - 0.931i)T \)
19 \( 1 + (-0.774 + 0.633i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.964 + 0.263i)T \)
31 \( 1 + (0.905 + 0.424i)T \)
37 \( 1 + (0.516 - 0.856i)T \)
41 \( 1 + (0.398 - 0.917i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (-0.217 + 0.976i)T \)
53 \( 1 + (-0.696 + 0.717i)T \)
59 \( 1 + (-0.595 + 0.803i)T \)
61 \( 1 + (0.749 - 0.662i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (0.897 - 0.441i)T \)
79 \( 1 + (0.179 + 0.983i)T \)
83 \( 1 + (-0.532 + 0.846i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.380 - 0.924i)T \)
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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

−18.49646834054090636059125630865, −17.64789921342859546104033896642, −17.05687915998589307864651187721, −16.21366120415593920979720840271, −15.40463349982715843151807769583, −14.92293618873772083940913076591, −14.681732981568506500283893131803, −13.337531444628317719265445596751, −12.96295064091629079391001510020, −12.241753620154706188630785110478, −11.389748080328562984500129125409, −10.893847938350257519518530353345, −10.01013250010985770697298018224, −9.60488748653511076683686236358, −8.44325028207963085387532863921, −8.077903059313388205273066235334, −7.10608039719939491463963522319, −6.40727899037701448073533564452, −5.91804900989420855080581070634, −4.96487509684368952418097368524, −4.07143767693693431637415807924, −3.094046677022798253504805247079, −2.815112302334292471697849967897, −1.84662950517151048474547347234, −0.44447583807712981168131814575, 0.6907699312279600563069386226, 1.50064616643533330697708614256, 2.54383568772823531744036280902, 3.5832458598761208805174119865, 4.13410214292011506409582472921, 4.85604814838818341285938088202, 5.62251745959999898865794150559, 6.57189680410109717536051845803, 7.271709331337728250610396458117, 7.814684436809232621664946689277, 8.833422808795794733054892588925, 9.32750433846492380363055798211, 9.94283659102949710232239774465, 10.90541907481388766729578557855, 11.48237842936901988519495681882, 12.47416311311781947867277895579, 12.65086519918836360291831494348, 13.67260085266731416668247696646, 14.06632257552486298649495551957, 14.99307559684371033120743481402, 15.86051240283773315718355062422, 16.29541775780469757153189731433, 17.0405308013264056069899059788, 17.23174699227906914068266181242, 18.49976984119936441634491115138

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