Properties

Label 2-21e2-441.122-c1-0-5
Degree $2$
Conductor $441$
Sign $-0.891 - 0.453i$
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  + (−1.75 + 1.88i)2-s + (0.0116 − 1.73i)3-s + (−0.346 − 4.61i)4-s + (−0.695 + 0.871i)5-s + (3.24 + 3.05i)6-s + (−2.60 − 0.480i)7-s + (5.29 + 4.22i)8-s + (−2.99 − 0.0402i)9-s + (−0.428 − 2.83i)10-s + (1.68 + 0.383i)11-s + (−8.00 + 0.545i)12-s + (2.33 − 2.52i)13-s + (5.46 − 4.07i)14-s + (1.50 + 1.21i)15-s + (−8.09 + 1.21i)16-s + (−0.251 + 3.34i)17-s + ⋯
L(s)  = 1  + (−1.23 + 1.33i)2-s + (0.00671 − 0.999i)3-s + (−0.173 − 2.30i)4-s + (−0.310 + 0.389i)5-s + (1.32 + 1.24i)6-s + (−0.983 − 0.181i)7-s + (1.87 + 1.49i)8-s + (−0.999 − 0.0134i)9-s + (−0.135 − 0.898i)10-s + (0.507 + 0.115i)11-s + (−2.31 + 0.157i)12-s + (0.648 − 0.699i)13-s + (1.46 − 1.08i)14-s + (0.387 + 0.313i)15-s + (−2.02 + 0.304i)16-s + (−0.0608 + 0.812i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.891 - 0.453i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0668239 + 0.278527i\)
\(L(\frac12)\) \(\approx\) \(0.0668239 + 0.278527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0116 + 1.73i)T \)
7 \( 1 + (2.60 + 0.480i)T \)
good2 \( 1 + (1.75 - 1.88i)T + (-0.149 - 1.99i)T^{2} \)
5 \( 1 + (0.695 - 0.871i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-1.68 - 0.383i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-2.33 + 2.52i)T + (-0.971 - 12.9i)T^{2} \)
17 \( 1 + (0.251 - 3.34i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (2.18 - 1.26i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.848 + 1.76i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (4.77 - 7.00i)T + (-10.5 - 26.9i)T^{2} \)
31 \( 1 + (9.00 - 5.20i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.28 - 1.55i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (-2.19 - 5.59i)T + (-30.0 + 27.8i)T^{2} \)
43 \( 1 + (-0.249 + 0.636i)T + (-31.5 - 29.2i)T^{2} \)
47 \( 1 + (-4.14 - 3.84i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (6.46 + 9.48i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (2.63 - 6.71i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-3.05 - 0.229i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (4.30 + 7.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.09 - 6.42i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (3.45 - 11.2i)T + (-60.3 - 41.1i)T^{2} \)
79 \( 1 + (-6.65 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.945 + 0.876i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (10.3 - 9.58i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-11.4 + 6.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.08151461814515250404433236066, −10.49671372978504970628654229440, −9.262565111127489827054776572544, −8.647429629923249644488381062188, −7.68955792628854656100611605863, −6.98525749467682769973802166346, −6.33606889824228520215033920083, −5.55595626510483271643412205983, −3.42123684088985320338219888852, −1.39260916804823898888362396352, 0.28487704784686768184293728888, 2.34523218572643386261203485267, 3.57695726599496061203996607195, 4.24704198103396789918237947440, 6.00033578665892560987015096289, 7.46275379998977436912508166937, 8.669033311882899208022213780674, 9.247116600581487981694690586889, 9.636562499859293894152238761108, 10.73853735623699834516497857916

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