Properties

Label 1-21e2-441.310-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.250 - 0.968i$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (0.623 + 0.781i)8-s + (0.365 + 0.930i)10-s + (−0.733 + 0.680i)11-s + (−0.733 + 0.680i)13-s + (0.623 − 0.781i)16-s + (0.826 − 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.826 − 0.563i)20-s + (0.826 + 0.563i)22-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (0.826 + 0.563i)26-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (0.623 + 0.781i)8-s + (0.365 + 0.930i)10-s + (−0.733 + 0.680i)11-s + (−0.733 + 0.680i)13-s + (0.623 − 0.781i)16-s + (0.826 − 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.826 − 0.563i)20-s + (0.826 + 0.563i)22-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (0.826 + 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.250 - 0.968i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.250 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4141391907 - 0.5347697966i\)
\(L(\frac12)\) \(\approx\) \(0.4141391907 - 0.5347697966i\)
\(L(1)\) \(\approx\) \(0.6152389038 - 0.3080677211i\)
\(L(1)\) \(\approx\) \(0.6152389038 - 0.3080677211i\)

Euler product

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

−24.30214385576468528072844785246, −23.42420620683457317567298695141, −23.07551806034818487801433686091, −21.9121048626936221124709064097, −20.95735994843889681712132841985, −19.586577724586649968449035688189, −19.18115622280331954364701821100, −18.21044323772873059902358188128, −17.22242820028621722182799308740, −16.39657960056838165584556400257, −15.66919236982597511558027219848, −14.92975588582778702561189408384, −14.06970621434136955881805303047, −12.90783849628213549279559016612, −12.17777454832494931713198086580, −10.78534615482459146285234655089, −10.01750667920704018255840441196, −8.73257418751333100293636910286, −7.929347183356142017968597678800, −7.44531077679490279904154842053, −6.06267324810311377284323709696, −5.23265683551928399755561075513, −4.15497202531513803591590615357, −3.0737665736557341200233428165, −0.99259855894190416178311265944, 0.546957046619162597229577705359, 2.26852708290320297829477949724, 3.07389753520746535278759950284, 4.42509533206697208980962236223, 4.88864356451844404096343402101, 6.80451615382582633017728318568, 7.76314445305826481700507408337, 8.57198665205045273358044689419, 9.74314785812930320191051078543, 10.47697531640474814004290129657, 11.54828341879290011489205015809, 12.13642688953587699693121415498, 12.96010267276111818955577888125, 14.10728879605569100230047343477, 15.00120168104857327769021182496, 16.05629661507873899368438379926, 17.045641374299531243892191079752, 17.98921974355351473062301056204, 18.9399182204121960336444252783, 19.39940934915288170464307540294, 20.38856875654967991116957405926, 21.04736246261139768430001070729, 22.04421486722564172861889040442, 22.95718375993152469880778091338, 23.45894331213773546846129722709

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