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Properties

Label 2-21e2-63.16-c1-0-8
Degree $2$
Conductor $441$
Sign $0.0977 - 0.995i$
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  + (0.849 + 1.47i)2-s + (−1.58 − 0.709i)3-s + (−0.444 + 0.769i)4-s − 0.949·5-s + (−0.298 − 2.92i)6-s + 1.88·8-s + (1.99 + 2.24i)9-s + (−0.806 − 1.39i)10-s − 0.588·11-s + (1.24 − 0.901i)12-s + (2.50 + 4.34i)13-s + (1.49 + 0.673i)15-s + (2.49 + 4.31i)16-s + (3.79 + 6.56i)17-s + (−1.60 + 4.83i)18-s + (2.23 − 3.86i)19-s + ⋯
L(s)  = 1  + (0.600 + 1.04i)2-s + (−0.912 − 0.409i)3-s + (−0.222 + 0.384i)4-s − 0.424·5-s + (−0.121 − 1.19i)6-s + 0.667·8-s + (0.664 + 0.747i)9-s + (−0.255 − 0.441i)10-s − 0.177·11-s + (0.360 − 0.260i)12-s + (0.696 + 1.20i)13-s + (0.387 + 0.173i)15-s + (0.623 + 1.07i)16-s + (0.919 + 1.59i)17-s + (−0.378 + 1.14i)18-s + (0.511 − 0.886i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0977 - 0.995i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09101 + 0.989089i\)
\(L(\frac12)\) \(\approx\) \(1.09101 + 0.989089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.58 + 0.709i)T \)
7 \( 1 \)
good2 \( 1 + (-0.849 - 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.949T + 5T^{2} \)
11 \( 1 + 0.588T + 11T^{2} \)
13 \( 1 + (-2.50 - 4.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.79 - 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (2.73 - 4.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.03 + 5.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.49 + 6.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.527 - 0.913i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.73 - 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.46 + 5.99i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.21 - 9.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.82 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.93 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + (2.23 + 3.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.666 - 1.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.84 + 4.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.421 - 0.730i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.70 + 2.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.26756004749611396598973439413, −10.80078649084929110335281296971, −9.514152148961519863469568021317, −8.085685284883118327273958952408, −7.43784661944065481283043432042, −6.44626811704803239477505969776, −5.88548142872285505644324061227, −4.81789017413554355518807265117, −3.89189199461276534433838663868, −1.54756287828522052360527471405, 1.01042773970483991256133649831, 3.02103317421056553031080166242, 3.83728805154576352174017860577, 5.00837443172295997938633620922, 5.72485369069554804080742510185, 7.21188722456482725787572391583, 8.065294704365261235268104795892, 9.677657721610664154604191235128, 10.28266831739139794080793961076, 11.10506294667768773092692892004

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