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Properties

Label 1-21e2-441.104-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: $\chi_{441} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06185276797 - 0.4541828610i\)
\(L(\frac12)\) \(\approx\) \(0.06185276797 - 0.4541828610i\)
\(L(1)\) \(\approx\) \(0.5361886327 - 0.2599175520i\)
\(L(1)\) \(\approx\) \(0.5361886327 - 0.2599175520i\)

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

−24.54147466752859835383569367391, −23.70639013077397147472034657726, −22.94194242515755045648820309129, −21.70549754563285163547702882998, −20.97975013576104532576451161341, −20.00309214265848343423626059824, −18.90943162843262121973448500785, −18.514116668190980243720010022760, −17.5926009498400037041389183909, −16.89934012391544442374865534569, −15.57018293439102987847157704146, −15.27942973008168768514013461434, −14.08244743493175736000163220586, −13.20361922918508228349009481681, −11.72929575492736139794632731546, −10.86685869748947507649974034332, −10.26429537619128292257986215713, −9.28074609061547269913034637385, −8.268580171093758015854473227660, −7.33036780608889825381306009083, −6.41058378798483783697551784191, −5.70576729780786296914884622113, −4.06276983224058405990245711916, −2.538662048097855709846954046992, −1.82263669352661209063964737251, 0.328335827955689598021947002118, 1.71300692968381275766306070880, 2.74480481538825197320211488464, 4.12693209747427216415608393161, 5.45595669116741515314446775071, 6.43341174818748614634794422032, 7.73930785544094425741524527428, 8.558195623868346843741697621497, 9.152713409884970321492493969704, 10.41864910693900991326299327585, 10.88975997366195693212185248236, 12.25037985017965644199722676049, 12.88231276891517877578045291168, 13.7882201651570994273225097739, 15.41644430552376020080713268117, 15.99546125588748247070451288714, 16.81535548738131450771560302594, 17.79591980369891142195698427418, 18.28454303577499298387431400578, 19.42988800612014042634186231766, 20.26065719820009101400632638728, 20.90687742450020779865861993913, 21.54991962540620777313807670531, 22.77855212610721794399098881155, 24.04171089490232832326521733469

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