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Properties

Label 1-444-444.419-r0-0-0
Degree $1$
Conductor $444$
Sign $-0.556 - 0.831i$
Analytic cond. $2.06192$
Root an. cond. $2.06192$
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  + (−0.766 + 0.642i)5-s + (−0.766 + 0.642i)7-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (0.939 − 0.342i)17-s + (−0.173 − 0.984i)19-s + (−0.5 − 0.866i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)29-s − 31-s + (0.173 − 0.984i)35-s + (0.939 + 0.342i)41-s − 43-s + (−0.5 − 0.866i)47-s + (0.173 − 0.984i)49-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)5-s + (−0.766 + 0.642i)7-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (0.939 − 0.342i)17-s + (−0.173 − 0.984i)19-s + (−0.5 − 0.866i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)29-s − 31-s + (0.173 − 0.984i)35-s + (0.939 + 0.342i)41-s − 43-s + (−0.5 − 0.866i)47-s + (0.173 − 0.984i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.556 - 0.831i$
Analytic conductor: \(2.06192\)
Root analytic conductor: \(2.06192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 444,\ (0:\ ),\ -0.556 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09180749305 - 0.1718789929i\)
\(L(\frac12)\) \(\approx\) \(0.09180749305 - 0.1718789929i\)
\(L(1)\) \(\approx\) \(0.6354947160 + 0.05871622886i\)
\(L(1)\) \(\approx\) \(0.6354947160 + 0.05871622886i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
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

−24.021672380124284619691563189245, −23.65276682043851829405351826293, −22.77339567536323038608097980294, −21.71579044479403611775182735645, −20.88797076897822988225307150055, −19.8496125444884308584291769004, −19.36638962303579724158669365759, −18.5215039617979880798311848624, −17.126331421785752997077973557556, −16.44226065967169846620516577761, −15.95384939863855234253305837535, −14.7133080138689030701628758465, −13.82345579522513416160788204798, −12.73888408747531620393382908380, −12.20039918349776648750561149955, −11.05088239882559051921345681651, −10.08916614663328449068310477297, −9.17065267155070264297973384719, −7.98986932113061888839962661650, −7.410878065271272369967509751513, −6.09014696163403127509760245537, −5.06676842317634988550057239228, −3.875149361890809817596303708282, −3.1583254762257786095818652214, −1.38510480068046066895472073412, 0.11115017920459496353491213480, 2.37393975818388661161883572668, 3.03593960695351744657036473838, 4.33366592949366370467571237468, 5.39920527402910570967056485759, 6.650907343889521584073713839494, 7.39900617527702682755613720993, 8.345926049953297784559132105058, 9.66618816093239486297352626280, 10.245479352542151606323136267484, 11.48431529022862558866825784490, 12.30480047764833708735743111316, 12.961580589852795072684630884803, 14.37620996461773838001399288032, 15.10522544823792524077548625744, 15.764086199567164409050588519647, 16.69128842453441771762297739519, 17.935421829534351412693846014768, 18.56901604198525542638476772051, 19.514901533619028755316078420423, 20.05245695973970199198349824043, 21.29551035991406336821910729012, 22.24319420142524071046437679406, 22.80494973762808161271362643779, 23.5803016327958858143159170105

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