Properties

Label 2-21e2-49.32-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.722 + 0.691i$
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  + (−0.0998 − 1.33i)2-s + (0.211 − 0.0319i)4-s + (−2.75 + 2.56i)5-s + (0.0649 − 2.64i)7-s + (−0.658 − 2.88i)8-s + (3.68 + 3.42i)10-s + (3.72 − 2.53i)11-s + (−0.0589 − 0.0283i)13-s + (−3.53 + 0.177i)14-s + (−3.36 + 1.03i)16-s + (−2.49 − 6.35i)17-s + (−1.99 − 3.45i)19-s + (−0.503 + 0.630i)20-s + (−3.75 − 4.70i)22-s + (−1.55 + 3.95i)23-s + ⋯
L(s)  = 1  + (−0.0706 − 0.942i)2-s + (0.105 − 0.0159i)4-s + (−1.23 + 1.14i)5-s + (0.0245 − 0.999i)7-s + (−0.232 − 1.01i)8-s + (1.16 + 1.08i)10-s + (1.12 − 0.765i)11-s + (−0.0163 − 0.00787i)13-s + (−0.943 + 0.0474i)14-s + (−0.842 + 0.259i)16-s + (−0.605 − 1.54i)17-s + (−0.457 − 0.792i)19-s + (−0.112 + 0.141i)20-s + (−0.800 − 1.00i)22-s + (−0.323 + 0.824i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399157 - 0.994983i\)
\(L(\frac12)\) \(\approx\) \(0.399157 - 0.994983i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0649 + 2.64i)T \)
good2 \( 1 + (0.0998 + 1.33i)T + (-1.97 + 0.298i)T^{2} \)
5 \( 1 + (2.75 - 2.56i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (-3.72 + 2.53i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (0.0589 + 0.0283i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.49 + 6.35i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (1.99 + 3.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.55 - 3.95i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (-5.59 + 7.02i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.05 - 3.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.91 - 0.288i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-1.34 - 5.88i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.175 - 0.770i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.0438 + 0.585i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-3.00 + 0.452i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-1.36 - 1.27i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-4.88 - 0.735i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (1.53 - 2.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.78 - 4.74i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.783 + 10.4i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-6.49 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.60 - 1.25i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-9.50 - 6.48i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 - 9.43T + 97T^{2} \)
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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

−11.11600674857620441243505128970, −10.20125307053422785324957637428, −9.251943863924464812110500439117, −7.904705231512276702731953696129, −6.94175430201640886683026065061, −6.54089170697738687219156891423, −4.36491773814548981572366414888, −3.59034846338220927120483558781, −2.69578470240148259850041063800, −0.69812101542231532706438932000, 1.90864828772534107486048554036, 3.88261155249398853159406229477, 4.80177411568067110877128603465, 5.96054991537318725533935583790, 6.80836927691918883312082339123, 7.969465455396667100821593405791, 8.548362164230571083246648610155, 9.102623976759939782516655972005, 10.67436181198003325689213349086, 11.77820914194257232674170473770

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