Properties

Label 1-21e2-441.106-r0-0-0
Degree $1$
Conductor $441$
Sign $0.963 - 0.267i$
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  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (0.365 + 0.930i)5-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)10-s + (0.955 − 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.365 − 0.930i)16-s + (−0.900 + 0.433i)17-s + 19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (0.826 − 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)26-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (0.365 + 0.930i)5-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)10-s + (0.955 − 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.365 − 0.930i)16-s + (−0.900 + 0.433i)17-s + 19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (0.826 − 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.645204498 - 0.3602364469i\)
\(L(\frac12)\) \(\approx\) \(2.645204498 - 0.3602364469i\)
\(L(1)\) \(\approx\) \(1.984422889 - 0.2132457285i\)
\(L(1)\) \(\approx\) \(1.984422889 - 0.2132457285i\)

Euler product

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

−24.392305189477037464320368506856, −23.237486532749561802862316966866, −22.32499922995705466232303194117, −21.69367727007351348511273982004, −20.74933036862699962933699345103, −20.08144058212921814382312347944, −19.252293250854018343126121428122, −17.56087513345829725531798726919, −17.12883539720352601979772205634, −16.15971414324125748451783643035, −15.4199638384728415771299697364, −14.24033511598283474857384888808, −13.71949165569639419827006535215, −12.67994390857024108312997798344, −11.97372084695052938878900480618, −11.183258118047738282554252566530, −9.613386693170559156436204423172, −8.93635483957897903751346196153, −7.586295635639645128584369003373, −6.75053178713184587606231700467, −5.67037358636726855509139571351, −4.74542243971459919096851125326, −4.03681018793791502938618317965, −2.581547993031027331866216902211, −1.46664225589919935197573349362, 1.43511919454120142192164028716, 2.708344021130874192555891229595, 3.41256031026404995484821159321, 4.66426094634763649235086285116, 5.70841374460662343040834976005, 6.66956934504342476667933691571, 7.30577763364025958548696424431, 8.9322049477589279577719710223, 10.12309672925470314325445261883, 10.79324119346454439169486047322, 11.70379502996169145178909805659, 12.60011432633696815336734539352, 13.63380400381314630938358859356, 14.362781444135161095534197472011, 15.00964985897359757237742173544, 15.923005942581676860245370661443, 17.11155566849029122194987171087, 18.01359470023711372681998750781, 19.15099020459206540096705989658, 19.77869351077211852809654702821, 20.68512763221644927776081293140, 21.90564133885991876841082270140, 22.12926564565891751179642106119, 22.89614006778486715451688056113, 23.94600920404687184768119620341

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