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Properties

Label 1-21e2-441.403-r0-0-0
Degree $1$
Conductor $441$
Sign $0.577 - 0.816i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.955 − 0.294i)5-s + (−0.222 + 0.974i)8-s + (−0.733 + 0.680i)10-s + (0.0747 − 0.997i)11-s + (0.0747 − 0.997i)13-s + (−0.222 − 0.974i)16-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.365 − 0.930i)20-s + (0.365 + 0.930i)22-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (0.365 + 0.930i)26-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.955 − 0.294i)5-s + (−0.222 + 0.974i)8-s + (−0.733 + 0.680i)10-s + (0.0747 − 0.997i)11-s + (0.0747 − 0.997i)13-s + (−0.222 − 0.974i)16-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.365 − 0.930i)20-s + (0.365 + 0.930i)22-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (0.365 + 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8438730498 - 0.4364312314i\)
\(L(\frac12)\) \(\approx\) \(0.8438730498 - 0.4364312314i\)
\(L(1)\) \(\approx\) \(0.8217347915 - 0.09232835457i\)
\(L(1)\) \(\approx\) \(0.8217347915 - 0.09232835457i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (-0.733 + 0.680i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.988 - 0.149i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + T \)
83 \( 1 + (0.0747 + 0.997i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
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

−24.40573806587537060273609050790, −23.37290213449243952719100606530, −22.02540872608613611550778251742, −21.58892760408674397268350565878, −20.72765364024177931491728540291, −19.810663842173183382346126737, −18.98019184946033885215744242842, −18.08211536586079929188183668483, −17.41501787901108233483398152397, −16.7636293177338160158768565011, −15.62747203601213936766742942338, −14.628797647979382432757450814315, −13.557234843657539010672277389416, −12.5792368988070763783846317289, −11.72907416118903559200090590942, −10.59017559899583845836230993392, −9.966790883731451542718845957242, −9.14705686060765318017693219797, −8.20861942590853220069103889197, −6.89565645905588575542784559988, −6.40274595943144943455913371596, −4.82113403710511217605971783823, −3.5151920128291161458852175900, −2.19509827514307010134302960861, −1.582052239179342259478586196627, 0.72772135875131218656905766564, 1.96579042715335255462769138921, 3.15018046952689543540017728125, 5.0111810307115088181140863947, 5.88067352137736608294261067492, 6.55051742831592272285723176478, 8.01778217007381002292571629544, 8.50374224890728036131695227479, 9.77180610555246360950828946574, 10.189594477502745304702916277598, 11.30821346046659364664391684174, 12.4035764306686707396684921609, 13.71133876171752294208128438596, 14.24791468087949989992850906649, 15.4856212386785612547804574974, 16.29240412803117262545236261186, 17.063962148874428610268716909203, 17.826940933567798041118402787833, 18.59751137216635732629443261496, 19.43781167288817867266313116356, 20.57550193695752875980365183730, 21.03042020605383651786358432207, 22.25525284609696745459651917254, 23.20868515897903357447540045451, 24.356235254471801407583843511586

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