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Properties

Label 2-21e2-7.2-c1-0-10
Degree $2$
Conductor $441$
Sign $0.991 + 0.126i$
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  + (−0.866 + 1.5i)2-s + (−0.5 − 0.866i)4-s + (1.73 − 3i)5-s − 1.73·8-s + (3 + 5.19i)10-s + (−1.73 − 3i)11-s + 2·13-s + (2.49 − 4.33i)16-s + (−1.73 − 3i)17-s + (2 − 3.46i)19-s − 3.46·20-s + 6·22-s + (1.73 − 3i)23-s + (−3.5 − 6.06i)25-s + (−1.73 + 3i)26-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (−0.250 − 0.433i)4-s + (0.774 − 1.34i)5-s − 0.612·8-s + (0.948 + 1.64i)10-s + (−0.522 − 0.904i)11-s + 0.554·13-s + (0.624 − 1.08i)16-s + (−0.420 − 0.727i)17-s + (0.458 − 0.794i)19-s − 0.774·20-s + 1.27·22-s + (0.361 − 0.625i)23-s + (−0.700 − 1.21i)25-s + (−0.339 + 0.588i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04087 - 0.0660541i\)
\(L(\frac12)\) \(\approx\) \(1.04087 - 0.0660541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.73 + 3i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.46 - 6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.73 + 3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−11.01032789922896324300908660310, −9.768988302378158257617624412363, −8.901289151620908951317373383751, −8.602083726209326506024976057020, −7.53241019937762652152513424655, −6.40141010493233504366606581153, −5.59467185202947558000675716199, −4.73923974954126445975856877768, −2.85636413025418105774924826161, −0.828840513664867935599289602033, 1.74670460510648390999590675763, 2.64925095498153368402024617772, 3.76463270586229244986735471166, 5.61293143883103867370667363314, 6.45518653816995207796827242672, 7.50958530219792359034937777842, 8.727011964416920478322256094674, 9.894407666243160427149782490803, 10.11947764587308644532347165234, 10.98887805739241215794199083103

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