Properties

Label 2-21e2-441.5-c1-0-27
Degree $2$
Conductor $441$
Sign $0.110 + 0.993i$
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  + (−2.15 + 1.71i)2-s + (−1.44 − 0.960i)3-s + (1.24 − 5.46i)4-s + (−1.16 − 0.797i)5-s + (4.75 − 0.406i)6-s + (1.71 − 2.01i)7-s + (4.31 + 8.95i)8-s + (1.15 + 2.76i)9-s + (3.89 − 0.291i)10-s + (0.0494 + 0.327i)11-s + (−7.04 + 6.67i)12-s + (−0.0685 − 0.454i)13-s + (−0.231 + 7.29i)14-s + (0.919 + 2.27i)15-s + (−14.5 − 7.02i)16-s + (1.77 − 1.64i)17-s + ⋯
L(s)  = 1  + (−1.52 + 1.21i)2-s + (−0.832 − 0.554i)3-s + (0.623 − 2.73i)4-s + (−0.523 − 0.356i)5-s + (1.94 − 0.165i)6-s + (0.647 − 0.761i)7-s + (1.52 + 3.16i)8-s + (0.384 + 0.923i)9-s + (1.23 − 0.0922i)10-s + (0.0149 + 0.0988i)11-s + (−2.03 + 1.92i)12-s + (−0.0189 − 0.126i)13-s + (−0.0617 + 1.94i)14-s + (0.237 + 0.586i)15-s + (−3.64 − 1.75i)16-s + (0.429 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.243581 - 0.218049i\)
\(L(\frac12)\) \(\approx\) \(0.243581 - 0.218049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.44 + 0.960i)T \)
7 \( 1 + (-1.71 + 2.01i)T \)
good2 \( 1 + (2.15 - 1.71i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (1.16 + 0.797i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.0494 - 0.327i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (0.0685 + 0.454i)T + (-12.4 + 3.83i)T^{2} \)
17 \( 1 + (-1.77 + 1.64i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-4.18 - 2.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.02 + 6.55i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (1.43 + 1.54i)T + (-2.16 + 28.9i)T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 + (8.11 + 2.50i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (0.100 - 1.33i)T + (-40.5 - 6.11i)T^{2} \)
43 \( 1 + (0.468 + 6.25i)T + (-42.5 + 6.40i)T^{2} \)
47 \( 1 + (-5.40 - 6.78i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.76 + 8.95i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (8.56 + 4.12i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-10.0 + 2.28i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 + (6.85 + 1.56i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.873 + 5.79i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 - 0.748T + 79T^{2} \)
83 \( 1 + (2.02 + 0.305i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (4.01 + 10.2i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (7.31 - 4.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.54160457281037046351167945609, −10.12251493723161794140280362006, −8.756725082960909226639395889192, −7.946810533123919546386630159426, −7.39784876786768576556238729022, −6.59621943062195957532921482392, −5.50634843800033679839744944569, −4.66842950445847014453300536014, −1.60487044097833608387710410193, −0.41693482487879501512372693696, 1.46483979960429791748880807134, 3.07765379698145830647120656999, 4.07900340098691454313980326772, 5.60977761446912086145390255798, 7.17265167693246562132248682124, 7.88490635432746681801358330110, 9.016638743299388728309770615395, 9.603442375533946271010347854919, 10.52784778475775723638213652506, 11.33162220567296587759624796219

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