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Properties

Label 2-21e2-63.5-c1-0-9
Degree $2$
Conductor $441$
Sign $-0.524 - 0.851i$
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  + 0.718i·2-s + (0.271 + 1.71i)3-s + 1.48·4-s + (0.723 + 1.25i)5-s + (−1.22 + 0.194i)6-s + 2.50i·8-s + (−2.85 + 0.928i)9-s + (−0.900 + 0.519i)10-s + (−1.55 − 0.900i)11-s + (0.402 + 2.53i)12-s + (1.88 + 1.09i)13-s + (−1.94 + 1.57i)15-s + 1.17·16-s + (−1.95 − 3.38i)17-s + (−0.667 − 2.04i)18-s + (3.47 + 2.00i)19-s + ⋯
L(s)  = 1  + 0.507i·2-s + (0.156 + 0.987i)3-s + 0.742·4-s + (0.323 + 0.560i)5-s + (−0.501 + 0.0795i)6-s + 0.884i·8-s + (−0.950 + 0.309i)9-s + (−0.284 + 0.164i)10-s + (−0.470 − 0.271i)11-s + (0.116 + 0.732i)12-s + (0.523 + 0.302i)13-s + (−0.502 + 0.407i)15-s + 0.292·16-s + (−0.473 − 0.820i)17-s + (−0.157 − 0.482i)18-s + (0.797 + 0.460i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863121 + 1.54646i\)
\(L(\frac12)\) \(\approx\) \(0.863121 + 1.54646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.271 - 1.71i)T \)
7 \( 1 \)
good2 \( 1 - 0.718iT - 2T^{2} \)
5 \( 1 + (-0.723 - 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.55 + 0.900i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.88 - 1.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.47 - 2.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.91 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.49 + 4.90i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 + (0.411 - 0.713i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.90 - 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.76 + 6.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.33T + 47T^{2} \)
53 \( 1 + (-0.996 + 0.575i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.79T + 59T^{2} \)
61 \( 1 + 2.35iT - 61T^{2} \)
67 \( 1 + 0.312T + 67T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (2.42 - 1.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (3.60 + 6.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.28 + 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 + 7.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.45918976785738509800060831127, −10.34091459476827179728370185943, −9.890658756049826295622353012359, −8.575272958190037818375406898554, −7.83367044291382765631884181668, −6.61118730823860971580639099885, −5.86509916600199491868463292350, −4.81970577853011079679272453201, −3.35937815628962683166431625128, −2.36398795213447887577533666323, 1.18102615293436640557812269561, 2.28371162782614783119735658250, 3.43932382490988152209665734474, 5.17610968614366160006583970836, 6.28676493720901408117413768524, 7.02512355860691926554920628802, 8.104417438403704514015101027951, 8.875203585026118889664731788482, 10.14239279856638837139793743740, 10.88584214775281307634850407735

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