Properties

Label 2-21e2-63.4-c1-0-18
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: $\chi_{441} (67, \cdot )$
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

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

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