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Properties

Label 2-21e2-7.6-c4-0-28
Degree $2$
Conductor $441$
Sign $0.755 - 0.654i$
Analytic cond. $45.5861$
Root an. cond. $6.75175$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.179·2-s − 15.9·4-s + 39.6i·5-s + 5.73·8-s − 7.11i·10-s + 51.6·11-s − 223. i·13-s + 254.·16-s − 491. i·17-s + 187. i·19-s − 633. i·20-s − 9.27·22-s + 91.8·23-s − 946.·25-s + 40.1i·26-s + ⋯
L(s)  = 1  − 0.0448·2-s − 0.997·4-s + 1.58i·5-s + 0.0896·8-s − 0.0711i·10-s + 0.427·11-s − 1.32i·13-s + 0.993·16-s − 1.70i·17-s + 0.519i·19-s − 1.58i·20-s − 0.0191·22-s + 0.173·23-s − 1.51·25-s + 0.0594i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(45.5861\)
Root analytic conductor: \(6.75175\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :2),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.453739928\)
\(L(\frac12)\) \(\approx\) \(1.453739928\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 0.179T + 16T^{2} \)
5 \( 1 - 39.6iT - 625T^{2} \)
11 \( 1 - 51.6T + 1.46e4T^{2} \)
13 \( 1 + 223. iT - 2.85e4T^{2} \)
17 \( 1 + 491. iT - 8.35e4T^{2} \)
19 \( 1 - 187. iT - 1.30e5T^{2} \)
23 \( 1 - 91.8T + 2.79e5T^{2} \)
29 \( 1 - 937.T + 7.07e5T^{2} \)
31 \( 1 - 589. iT - 9.23e5T^{2} \)
37 \( 1 - 434.T + 1.87e6T^{2} \)
41 \( 1 - 694. iT - 2.82e6T^{2} \)
43 \( 1 + 933.T + 3.41e6T^{2} \)
47 \( 1 - 50.2iT - 4.87e6T^{2} \)
53 \( 1 - 3.37e3T + 7.89e6T^{2} \)
59 \( 1 - 3.45e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.25e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.62e3T + 2.01e7T^{2} \)
71 \( 1 + 1.79e3T + 2.54e7T^{2} \)
73 \( 1 + 5.10e3iT - 2.83e7T^{2} \)
79 \( 1 - 965.T + 3.89e7T^{2} \)
83 \( 1 + 4.45e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.52e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.05e4iT - 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35230470754922909441504558324, −9.957994208739013996483321045186, −8.837472812953380205909262496981, −7.79157495240375845461453508649, −6.98800390981726758293750552529, −5.89214985224559502607815411567, −4.83073528031723587824786219294, −3.49011550908046115704542046655, −2.75181628412003468266991495172, −0.74598429940511229108929613989, 0.68438539719341961350711558836, 1.72793702004739845636951514962, 3.91530446667762353202296032074, 4.47896677820997572803924783895, 5.41184095418318085712581987559, 6.54855544408517132823906842289, 8.086324377433695600822409816102, 8.696588221819706077944321501300, 9.270270731936867583118084049742, 10.12322016448221252752527653880

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