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Properties

Label 1-21e2-441.416-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.904 + 0.427i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (−0.826 + 0.563i)11-s + (−0.826 + 0.563i)13-s + (−0.222 + 0.974i)16-s + (−0.988 − 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (−0.826 + 0.563i)11-s + (−0.826 + 0.563i)13-s + (−0.222 + 0.974i)16-s + (−0.988 − 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.904 + 0.427i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.904 + 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3087489124 + 1.375279378i\)
\(L(\frac12)\) \(\approx\) \(0.3087489124 + 1.375279378i\)
\(L(1)\) \(\approx\) \(1.060444819 + 0.7678082088i\)
\(L(1)\) \(\approx\) \(1.060444819 + 0.7678082088i\)

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 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 - T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
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 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
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

−23.62360893348974132398343000026, −22.84434927371497790322549338775, −22.034253724860520770186845169442, −21.0761913657415175259478974151, −20.3231672115828291775142172993, −19.67066572142649074008268864717, −18.83651324340418634759091088326, −17.701847749207980246140376405072, −16.248962286789626775257425960798, −15.93119990493690431181248693993, −14.85750081828704610333180617127, −14.02147599304958033837446468462, −12.75928838521927594683591349508, −12.57228833648632988611715352616, −11.373558404370038030447736863597, −10.64534377069420167077081668611, −9.55237058885874438275256269753, −8.26342287991494170155886086463, −7.400311123849923911907335666, −6.06890551408956922373571554728, −5.11948577933421184234925817975, −4.31697399733701339551468888155, −3.2463922876116305245289851851, −2.150721952368702519182606096871, −0.55159520863740881381858723260, 2.25235784193731467776738006939, 3.05647684055077290133861913797, 4.30607393342085758848708202482, 4.98124660305006963932514457787, 6.3265071478813749961664381572, 7.26808123783545188416905451186, 7.725871422293715532353706038570, 9.1121054215237144492368377455, 10.48650806233287669418457820008, 11.41070080972349928299455204385, 12.098985225038888652070226181109, 13.13072341883703718333693168104, 14.000609867397611571056932738431, 14.94138215946122117500660974436, 15.56354114367197812032204911, 16.2517962380770252470515797237, 17.518092264755614611039272632467, 18.20535850353438345072937419017, 19.60134958895551122234482533196, 20.06458288655732786546271461868, 21.29915226169653124259054337317, 22.05988087938019789096442760256, 22.69789055647153033560649899245, 23.73289968750472815673222250230, 24.014657348491473750953833519499

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