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Properties

Label 2-21e2-63.4-c1-0-24
Degree $2$
Conductor $441$
Sign $-0.404 + 0.914i$
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  + (−1.02 + 1.77i)2-s + (−1.09 − 1.33i)3-s + (−1.10 − 1.92i)4-s + 0.146·5-s + (3.50 − 0.582i)6-s + 0.446·8-s + (−0.580 + 2.94i)9-s + (−0.150 + 0.260i)10-s + 1.66·11-s + (−1.34 + 3.59i)12-s + (−0.0999 + 0.173i)13-s + (−0.160 − 0.195i)15-s + (1.75 − 3.04i)16-s + (−3.13 + 5.43i)17-s + (−4.63 − 4.05i)18-s + (−3.45 − 5.99i)19-s + ⋯
L(s)  = 1  + (−0.726 + 1.25i)2-s + (−0.635 − 0.772i)3-s + (−0.554 − 0.960i)4-s + 0.0654·5-s + (1.43 − 0.237i)6-s + 0.157·8-s + (−0.193 + 0.981i)9-s + (−0.0474 + 0.0822i)10-s + 0.501·11-s + (−0.389 + 1.03i)12-s + (−0.0277 + 0.0480i)13-s + (−0.0415 − 0.0505i)15-s + (0.439 − 0.761i)16-s + (−0.760 + 1.31i)17-s + (−1.09 − 0.955i)18-s + (−0.793 − 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.404 + 0.914i$
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: \((2,\ 441,\ (\ :1/2),\ -0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0350439 - 0.0538494i\)
\(L(\frac12)\) \(\approx\) \(0.0350439 - 0.0538494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.09 + 1.33i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 - 1.77i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.146T + 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 + (0.0999 - 0.173i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.13 - 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 + (2.46 + 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.15 - 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.940 + 1.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.905 - 1.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.67 - 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.28 + 3.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.339 - 0.587i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + (-0.778 + 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.39 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.75 + 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.53 + 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.98 - 6.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.81352108173008308059999233466, −9.695325482122762267456729410775, −8.680711421431440680839101441258, −7.981223253586774784737198332027, −7.06023110783889630487823343478, −6.29837106996329466102027198090, −5.71257409296648222780415222609, −4.25814075098477092508009064757, −2.04805972593632103618272941636, −0.05292407288632756043181112424, 1.79940533187213069764868299288, 3.34583543579756330951173472823, 4.27834048765938654726977640494, 5.63575368709852133468571654147, 6.65055777808006510859257919287, 8.213331237082488968306535946593, 9.137929540374334489140348810265, 9.842007217082806416824111205509, 10.44513132887475999232646924466, 11.32542100959882832204182018054

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