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Properties

Label 2-21e2-49.15-c1-0-8
Degree $2$
Conductor $441$
Sign $-0.122 - 0.992i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.271 + 1.19i)2-s + (0.457 + 0.220i)4-s + (0.164 − 0.205i)5-s + (2.60 + 0.471i)7-s + (−1.91 + 2.39i)8-s + (0.200 + 0.251i)10-s + (−0.153 + 0.674i)11-s + (−0.135 + 0.592i)13-s + (−1.26 + 2.97i)14-s + (−1.69 − 2.13i)16-s + (3.45 − 1.66i)17-s − 0.615·19-s + (0.120 − 0.0580i)20-s + (−0.761 − 0.366i)22-s + (5.76 + 2.77i)23-s + ⋯
L(s)  = 1  + (−0.192 + 0.842i)2-s + (0.228 + 0.110i)4-s + (0.0734 − 0.0920i)5-s + (0.984 + 0.178i)7-s + (−0.675 + 0.846i)8-s + (0.0634 + 0.0795i)10-s + (−0.0464 + 0.203i)11-s + (−0.0375 + 0.164i)13-s + (−0.339 + 0.794i)14-s + (−0.424 − 0.532i)16-s + (0.837 − 0.403i)17-s − 0.141·19-s + (0.0269 − 0.0129i)20-s + (−0.162 − 0.0781i)22-s + (1.20 + 0.578i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.122 - 0.992i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.122 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01610 + 1.14915i\)
\(L(\frac12)\) \(\approx\) \(1.01610 + 1.14915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.471i)T \)
good2 \( 1 + (0.271 - 1.19i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (-0.164 + 0.205i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (0.153 - 0.674i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.135 - 0.592i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-3.45 + 1.66i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 0.615T + 19T^{2} \)
23 \( 1 + (-5.76 - 2.77i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (1.87 - 0.904i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 5.96T + 31T^{2} \)
37 \( 1 + (2.85 - 1.37i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (0.738 - 0.925i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (5.16 + 6.48i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.14 + 9.38i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (1.39 + 0.669i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-6.23 - 7.82i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-7.51 + 3.61i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + (4.67 + 2.24i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (2.50 + 10.9i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + (-1.83 - 8.05i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (4.13 + 18.0i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−11.45380499474327799140314182544, −10.54282990644287894540209297437, −9.212224834648531426957778234790, −8.550116247356888655591399632284, −7.49218650829540965177106027525, −7.03471946879738093057574839325, −5.60853252268911340173039228421, −5.04939410081506688566142806625, −3.34334037463718091472534801107, −1.82871402285270807530788152037, 1.16076739938975740007815714324, 2.44224693640540609468530683664, 3.67188939507349576150391075791, 5.01240984869369901350104842799, 6.15135685461276411730771150575, 7.23999218180039598934729414259, 8.261009265007019627909756366261, 9.245465018318986891184798038928, 10.31075751073534637466115795797, 10.84453885394597505643628409308

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