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Properties

Label 1-21e2-441.211-r0-0-0
Degree $1$
Conductor $441$
Sign $0.417 - 0.908i$
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  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.733 + 0.680i)5-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.826 − 0.563i)11-s + (0.0747 + 0.997i)13-s + (−0.733 − 0.680i)16-s + (0.623 − 0.781i)17-s + 19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.733 + 0.680i)5-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.826 − 0.563i)11-s + (0.0747 + 0.997i)13-s + (−0.733 − 0.680i)16-s + (0.623 − 0.781i)17-s + 19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.676542287 - 1.074382474i\)
\(L(\frac12)\) \(\approx\) \(1.676542287 - 1.074382474i\)
\(L(1)\) \(\approx\) \(1.465862105 - 0.5540779126i\)
\(L(1)\) \(\approx\) \(1.465862105 - 0.5540779126i\)

Euler product

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

−24.09314740734504601895391825975, −23.38045253481539644058595759156, −22.689230959333969372249063080821, −21.84503490532467357014402352083, −20.78127010985973345660038615910, −20.12270534128603140529011045372, −19.332433506910202910476228697779, −17.824425707218885894766321910864, −17.10710144320970149156761922898, −16.26259427238768971751504125323, −15.38479189703208265111476017335, −14.82292872133431231323563300772, −13.67180397219347546154720730426, −12.7850723234100963065778903819, −12.0648247943241060764297798939, −11.35618332060132767417941330537, −9.89329121424410951497180231897, −8.657888245756760745562956250881, −7.845889119428439284693573579053, −7.05244972316248989780595589259, −5.78078698809848527468685061642, −4.973776415025015398949312802454, −3.92154144915499406116885901288, −3.15832643637450484032869731156, −1.38826369413143759887406892959, 1.03060564945187868068069480894, 2.53437570385384971068710886668, 3.49078789570785268849981010333, 4.26590522518396230659548334430, 5.4643792430755004005992371117, 6.62902598269171232291292582431, 7.27636963877607529061995846197, 8.80526380469740712160789306375, 9.83095336478668406821820875800, 10.915132300463475098596520807467, 11.65532196241888587532735207025, 12.14883847848917176876129171710, 13.51249396527542935369963850184, 14.30669738752735998984683857927, 14.8236707899146704491866888212, 16.02701070566997992148469459910, 16.61611415770275033249198219744, 18.44539248409374236645883661790, 18.7350736017871442271101652301, 19.78390254085368352583562063068, 20.40002309756522034748658289883, 21.570912903425763193646083026035, 22.18095275299961300144926134472, 22.9481430556792267709556774595, 23.71207045761721615650493789591

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