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Properties

Label 1-21e2-441.340-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: $\chi_{441} (340, \cdot )$
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

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

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