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Properties

Label 2-21e2-63.4-c1-0-27
Degree $2$
Conductor $441$
Sign $0.502 + 0.864i$
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  + (0.920 − 1.59i)2-s + (1.58 + 0.691i)3-s + (−0.695 − 1.20i)4-s − 1.33·5-s + (2.56 − 1.89i)6-s + 1.12·8-s + (2.04 + 2.19i)9-s + (−1.22 + 2.12i)10-s + 1.51·11-s + (−0.271 − 2.39i)12-s + (2.58 − 4.48i)13-s + (−2.11 − 0.923i)15-s + (2.42 − 4.19i)16-s + (−0.774 + 1.34i)17-s + (5.38 − 1.23i)18-s + (1.25 + 2.16i)19-s + ⋯
L(s)  = 1  + (0.650 − 1.12i)2-s + (0.916 + 0.399i)3-s + (−0.347 − 0.601i)4-s − 0.596·5-s + (1.04 − 0.773i)6-s + 0.396·8-s + (0.681 + 0.732i)9-s + (−0.388 + 0.673i)10-s + 0.456·11-s + (−0.0782 − 0.690i)12-s + (0.717 − 1.24i)13-s + (−0.547 − 0.238i)15-s + (0.605 − 1.04i)16-s + (−0.187 + 0.325i)17-s + (1.26 − 0.291i)18-s + (0.287 + 0.497i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.502 + 0.864i$
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: \((2,\ 441,\ (\ :1/2),\ 0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21476 - 1.27364i\)
\(L(\frac12)\) \(\approx\) \(2.21476 - 1.27364i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 - 0.691i)T \)
7 \( 1 \)
good2 \( 1 + (-0.920 + 1.59i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
11 \( 1 - 1.51T + 11T^{2} \)
13 \( 1 + (-2.58 + 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 + (0.0309 + 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.92 + 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.75 - 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.755 + 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.22 + 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.61 + 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (-1.37 + 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.80 + 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.703 + 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.09 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.03483514423804225298472100532, −10.23136302391429784547054680186, −9.474113904203885674076264748632, −8.045726550561063517039244192085, −7.81019147584096170087127700638, −6.01178638763030385803497229251, −4.58106007426927281003377283841, −3.76967414024611202620764028512, −3.07912170280033526303320125733, −1.68505845760428203374217051414, 1.84390941972700604460180257677, 3.72252134113733103744928619531, 4.30604828376719266830605054973, 5.78893590918157855457841565579, 6.83788014828454802105853902836, 7.31344097473194406127492220885, 8.364769968244112286055262976425, 9.027430865879565693284981043878, 10.24402375392758827034874550188, 11.57862115568768003898994416831

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