Properties

Label 2-21e2-21.5-c1-0-12
Degree $2$
Conductor $441$
Sign $0.851 + 0.524i$
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  + (1.22 + 0.707i)2-s + (1.22 − 2.12i)5-s − 2.82i·8-s + (3 − 1.73i)10-s + (−1.22 + 0.707i)11-s − 5.19i·13-s + (2.00 − 3.46i)16-s + (2.44 + 4.24i)17-s + (−1.5 − 0.866i)19-s − 2·22-s + (4.89 + 2.82i)23-s + (−0.499 − 0.866i)25-s + (3.67 − 6.36i)26-s + 2.82i·29-s + (−1.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.547 − 0.948i)5-s − 0.999i·8-s + (0.948 − 0.547i)10-s + (−0.369 + 0.213i)11-s − 1.44i·13-s + (0.500 − 0.866i)16-s + (0.594 + 1.02i)17-s + (−0.344 − 0.198i)19-s − 0.426·22-s + (1.02 + 0.589i)23-s + (−0.0999 − 0.173i)25-s + (0.720 − 1.24i)26-s + 0.525i·29-s + (−0.269 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09366 - 0.593103i\)
\(L(\frac12)\) \(\approx\) \(2.09366 - 0.593103i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-1.22 - 0.707i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (6.12 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 1.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 + (2.44 - 4.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−10.95740781313115726416690689642, −10.09697957904451925646223614728, −9.273117930740872840360001183443, −8.240628022079310673514498806300, −7.19803803538844416118098401750, −5.87888561710092322925668569894, −5.41715552751071598966957061792, −4.51446094399531861215859291780, −3.19440087598142994601714977897, −1.19273837477788576291842007937, 2.20184535836084874211972049644, 3.07071901464324400837511754963, 4.26747934783780615839524012952, 5.29657453434507699830714010670, 6.41021049900033197763884062021, 7.31882673586797768926991150514, 8.554350266020722067045081276386, 9.549128844595081353795185261676, 10.56244658458906775631458768660, 11.36407183170941596287492976876

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