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Properties

Label 2-21e2-9.2-c2-0-34
Degree $2$
Conductor $441$
Sign $0.416 - 0.909i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 + 0.645i)2-s + (2.77 + 1.13i)3-s + (−1.16 − 2.02i)4-s + (−4.18 + 2.41i)5-s + (2.36 + 3.06i)6-s − 8.17i·8-s + (6.40 + 6.32i)9-s − 6.24·10-s + (11.8 + 6.84i)11-s + (−0.937 − 6.93i)12-s + (8.59 + 14.8i)13-s + (−14.3 + 1.94i)15-s + (0.609 − 1.05i)16-s + 8.21i·17-s + (3.08 + 11.2i)18-s + 13.8·19-s + ⋯
L(s)  = 1  + (0.558 + 0.322i)2-s + (0.925 + 0.379i)3-s + (−0.291 − 0.505i)4-s + (−0.837 + 0.483i)5-s + (0.394 + 0.510i)6-s − 1.02i·8-s + (0.711 + 0.702i)9-s − 0.624·10-s + (1.07 + 0.621i)11-s + (−0.0781 − 0.578i)12-s + (0.661 + 1.14i)13-s + (−0.958 + 0.129i)15-s + (0.0381 − 0.0660i)16-s + 0.483i·17-s + (0.171 + 0.622i)18-s + 0.729·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.416 - 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.27691 + 1.46158i\)
\(L(\frac12)\) \(\approx\) \(2.27691 + 1.46158i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.77 - 1.13i)T \)
7 \( 1 \)
good2 \( 1 + (-1.11 - 0.645i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (4.18 - 2.41i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-11.8 - 6.84i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.59 - 14.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 8.21iT - 289T^{2} \)
19 \( 1 - 13.8T + 361T^{2} \)
23 \( 1 + (-16.0 + 9.26i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (18.1 + 10.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-25.7 - 44.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 27.6T + 1.36e3T^{2} \)
41 \( 1 + (-18.4 + 10.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-18.1 + 31.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (51.4 + 29.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 93.6iT - 2.80e3T^{2} \)
59 \( 1 + (-1.45 + 0.838i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-32.7 + 56.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.6 - 47.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 14.2iT - 5.04e3T^{2} \)
73 \( 1 + 65.3T + 5.32e3T^{2} \)
79 \( 1 + (-3.35 + 5.80i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-11.6 - 6.69i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 15.8iT - 7.92e3T^{2} \)
97 \( 1 + (40.7 - 70.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.04337160217026028572688260080, −10.02485653322745498307718418225, −9.239142424512684644444953386515, −8.469022188108446515203642806317, −7.12532067715638294998938289581, −6.63416770550470956309166119387, −5.05276886447841510319356197609, −4.05033813496377478861170211813, −3.51593593330666655240321028640, −1.60041723730684402309789664212, 1.00218103876103759331802866858, 2.92387579210259667240894547787, 3.63184715562200674655951385007, 4.50502123983410696740077032038, 5.90414642231805528624359516097, 7.39339140897793279625738065656, 8.061514896317876128642687453027, 8.764197259720797360559245689945, 9.548473838816242504964308958132, 11.15306651279982722501992439309

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