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Properties

Label 2-21e2-49.16-c1-0-6
Degree $2$
Conductor $441$
Sign $0.794 + 0.607i$
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  + (−0.687 − 1.75i)2-s + (−1.12 + 1.04i)4-s + (−2.01 + 1.37i)5-s + (−0.492 + 2.59i)7-s + (−0.781 − 0.376i)8-s + (3.79 + 2.58i)10-s + (3.09 + 0.465i)11-s + (1.91 − 2.39i)13-s + (4.89 − 0.923i)14-s + (−0.351 + 4.69i)16-s + (4.75 − 1.46i)17-s + (1.89 − 3.27i)19-s + (0.835 − 3.66i)20-s + (−1.30 − 5.73i)22-s + (7.98 + 2.46i)23-s + ⋯
L(s)  = 1  + (−0.485 − 1.23i)2-s + (−0.564 + 0.523i)4-s + (−0.901 + 0.614i)5-s + (−0.186 + 0.982i)7-s + (−0.276 − 0.133i)8-s + (1.19 + 0.817i)10-s + (0.931 + 0.140i)11-s + (0.529 − 0.664i)13-s + (1.30 − 0.246i)14-s + (−0.0879 + 1.17i)16-s + (1.15 − 0.355i)17-s + (0.433 − 0.751i)19-s + (0.186 − 0.818i)20-s + (−0.278 − 1.22i)22-s + (1.66 + 0.513i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891934 - 0.301854i\)
\(L(\frac12)\) \(\approx\) \(0.891934 - 0.301854i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.492 - 2.59i)T \)
good2 \( 1 + (0.687 + 1.75i)T + (-1.46 + 1.36i)T^{2} \)
5 \( 1 + (2.01 - 1.37i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-3.09 - 0.465i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (-1.91 + 2.39i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-4.75 + 1.46i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-1.89 + 3.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.98 - 2.46i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (1.97 - 8.67i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.56 - 2.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.35 - 4.97i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (3.91 + 1.88i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.73 - 1.79i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.640 - 1.63i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (0.452 - 0.419i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-2.44 - 1.67i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (5.95 + 5.52i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-3.57 - 6.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.22 + 9.76i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.749 + 1.90i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (2.34 - 4.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.62 + 9.55i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.10 + 0.316i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 19.4T + 97T^{2} \)
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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

−11.26740120027455811431206875374, −10.27189351913101904492476145426, −9.287734711728665137680847903074, −8.716781657015027898096138729555, −7.46661138881444501003988410910, −6.44855553724488907836818945711, −5.12981078092423494881849716440, −3.34122533010736991973973016685, −3.09169249257709322454251841938, −1.24723740692008032088753239556, 0.878876520758407395523239343469, 3.55087121458522014121490792059, 4.45361013415332819930777502089, 5.84362399079356246327616297465, 6.74741023721685345936016804594, 7.60841204467610019383976989222, 8.238343368854916560475058595121, 9.126623867058808882655625179579, 9.996226186031656141779304366316, 11.36339536957378123825212829397

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