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Properties

Label 1-66e2-4356.911-r0-0-0
Degree $1$
Conductor $4356$
Sign $0.989 + 0.143i$
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  + (0.683 − 0.730i)5-s + (−0.953 − 0.299i)7-s + (0.548 − 0.836i)13-s + (0.466 + 0.884i)17-s + (0.985 + 0.170i)19-s + (0.580 + 0.814i)23-s + (−0.0665 − 0.997i)25-s + (0.830 + 0.556i)29-s + (−0.988 − 0.151i)31-s + (−0.870 + 0.491i)35-s + (−0.254 + 0.967i)37-s + (0.851 + 0.524i)41-s + (−0.981 − 0.189i)43-s + (0.997 + 0.0760i)47-s + (0.820 + 0.572i)49-s + ⋯
L(s)  = 1  + (0.683 − 0.730i)5-s + (−0.953 − 0.299i)7-s + (0.548 − 0.836i)13-s + (0.466 + 0.884i)17-s + (0.985 + 0.170i)19-s + (0.580 + 0.814i)23-s + (−0.0665 − 0.997i)25-s + (0.830 + 0.556i)29-s + (−0.988 − 0.151i)31-s + (−0.870 + 0.491i)35-s + (−0.254 + 0.967i)37-s + (0.851 + 0.524i)41-s + (−0.981 − 0.189i)43-s + (0.997 + 0.0760i)47-s + (0.820 + 0.572i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896847241 + 0.1367652568i\)
\(L(\frac12)\) \(\approx\) \(1.896847241 + 0.1367652568i\)
\(L(1)\) \(\approx\) \(1.178340461 - 0.09089180607i\)
\(L(1)\) \(\approx\) \(1.178340461 - 0.09089180607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.683 - 0.730i)T \)
7 \( 1 + (-0.953 - 0.299i)T \)
13 \( 1 + (0.548 - 0.836i)T \)
17 \( 1 + (0.466 + 0.884i)T \)
19 \( 1 + (0.985 + 0.170i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.830 + 0.556i)T \)
31 \( 1 + (-0.988 - 0.151i)T \)
37 \( 1 + (-0.254 + 0.967i)T \)
41 \( 1 + (0.851 + 0.524i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (0.997 + 0.0760i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (0.879 + 0.475i)T \)
61 \( 1 + (-0.761 - 0.647i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (0.993 + 0.113i)T \)
79 \( 1 + (-0.820 + 0.572i)T \)
83 \( 1 + (0.00951 + 0.999i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.683 - 0.730i)T \)
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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

−18.2433410460972978013545371582, −17.89257923481871281205499762637, −16.80333578631483143679526364133, −16.303299652814638332526729708694, −15.70902845662095335080396668424, −14.87334039973479047919895129358, −14.07257968070381728120014672327, −13.745497995340423254263811816690, −12.93968093784565237589514307883, −12.18633260432849212978705199328, −11.46574989003467111299819582837, −10.72428895619675014413365598783, −10.04021627014932238529887868409, −9.26906548343787166715505934458, −9.03250101968745939531513295133, −7.78426539104148502516560414064, −6.92492112481650347850319806790, −6.59369035347422435411173945720, −5.72743205573546531004554948244, −5.1346339978679867761335656392, −3.98257700908285650997372302705, −3.20398519014047851145890888771, −2.62417145488289115623641524202, −1.79019163724737066086482313520, −0.62035094252817462447580403593, 0.96897366272776722060499586900, 1.396578730020483379144304349561, 2.68804200360599750674473647152, 3.36481123365938348694042863871, 4.08625664205963924030687129203, 5.20662110604394905172879740632, 5.65912330094824713941595883406, 6.35138768234344747446126236954, 7.19831029838291659151417348017, 8.025297497294056318600014094361, 8.75348346961880429846423410674, 9.46798232824064125519271067472, 10.05724860794621437969471574371, 10.61542265238301840942162475517, 11.5583883159254890980621686805, 12.611818667022506947983239226003, 12.73540689288800760735210460674, 13.609962407653841921929388603139, 14.01179601479530628235378443828, 15.0935714782980594102345658922, 15.72488462681151867077989808707, 16.39808422911371726666347582998, 16.93671098990817605516950410022, 17.57647612545336400238760791067, 18.28925005108607587318870264632

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