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Properties

Label 2-21e2-63.16-c1-0-9
Degree $2$
Conductor $441$
Sign $0.940 + 0.339i$
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  + (−1.23 − 2.13i)2-s + (0.410 + 1.68i)3-s + (−2.02 + 3.51i)4-s + 3.65·5-s + (3.08 − 2.94i)6-s + 5.05·8-s + (−2.66 + 1.38i)9-s + (−4.50 − 7.79i)10-s + 0.406·11-s + (−6.73 − 1.97i)12-s + (−0.243 − 0.421i)13-s + (1.5 + 6.15i)15-s + (−2.16 − 3.74i)16-s + (2.42 + 4.20i)17-s + (6.21 + 3.97i)18-s + (−0.986 + 1.70i)19-s + ⋯
L(s)  = 1  + (−0.869 − 1.50i)2-s + (0.236 + 0.971i)3-s + (−1.01 + 1.75i)4-s + 1.63·5-s + (1.25 − 1.20i)6-s + 1.78·8-s + (−0.887 + 0.460i)9-s + (−1.42 − 2.46i)10-s + 0.122·11-s + (−1.94 − 0.569i)12-s + (−0.0675 − 0.116i)13-s + (0.387 + 1.58i)15-s + (−0.540 − 0.936i)16-s + (0.588 + 1.01i)17-s + (1.46 + 0.937i)18-s + (−0.226 + 0.392i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.940 + 0.339i$
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: \((2,\ 441,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14959 - 0.201395i\)
\(L(\frac12)\) \(\approx\) \(1.14959 - 0.201395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.410 - 1.68i)T \)
7 \( 1 \)
good2 \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.65T + 5T^{2} \)
11 \( 1 - 0.406T + 11T^{2} \)
13 \( 1 + (0.243 + 0.421i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.42 - 4.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.986 - 1.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.64T + 23T^{2} \)
29 \( 1 + (3.82 - 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.51 + 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.16 - 2.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.16 + 2.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.15 + 5.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.78 - 3.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.05 + 5.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.01 + 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.80 - 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (-0.986 - 1.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.08 + 7.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.08 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.41 + 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.74 - 8.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63646881848987621458039972157, −10.24260999674362066825781690444, −9.481473827915657419005228065410, −8.966447268138470369251690335363, −8.009540173501288924978581995054, −6.17746661612132195175823685110, −5.09229647410518849255067589246, −3.70877599769213204040028168154, −2.67652209106544895360883273411, −1.58851712260859333159429928568, 1.09089056097288579475028094073, 2.57018625239071644624316434413, 5.15857583633516194606891164798, 5.86334149062694961544567358485, 6.68983603434044607714415402687, 7.30774058688353039201439720642, 8.373851796510073132684942357676, 9.271612774345158105416752311606, 9.638168751219210089804826469206, 10.85179327175889160553685633679

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