Properties

Label 2-21e2-147.104-c1-0-12
Degree $2$
Conductor $441$
Sign $0.0164 - 0.999i$
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  + (1.61 + 1.28i)2-s + (0.505 + 2.21i)4-s + (3.93 + 1.89i)5-s + (−2.02 + 1.70i)7-s + (−0.245 + 0.509i)8-s + (3.91 + 8.12i)10-s + (−3.17 − 2.53i)11-s + (−1.07 − 0.858i)13-s + (−5.46 + 0.147i)14-s + (3.04 − 1.46i)16-s + (−0.000551 + 0.00241i)17-s − 4.84i·19-s + (−2.20 + 9.67i)20-s + (−1.87 − 8.19i)22-s + (−1.08 + 0.246i)23-s + ⋯
L(s)  = 1  + (1.14 + 0.911i)2-s + (0.252 + 1.10i)4-s + (1.75 + 0.846i)5-s + (−0.764 + 0.644i)7-s + (−0.0867 + 0.180i)8-s + (1.23 + 2.56i)10-s + (−0.958 − 0.764i)11-s + (−0.298 − 0.238i)13-s + (−1.46 + 0.0395i)14-s + (0.760 − 0.366i)16-s + (−0.000133 + 0.000585i)17-s − 1.11i·19-s + (−0.493 + 2.16i)20-s + (−0.398 − 1.74i)22-s + (−0.225 + 0.0514i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0164 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0164 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0164 - 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0164 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05908 + 2.02550i\)
\(L(\frac12)\) \(\approx\) \(2.05908 + 2.02550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 - 1.70i)T \)
good2 \( 1 + (-1.61 - 1.28i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-3.93 - 1.89i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (3.17 + 2.53i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.07 + 0.858i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.000551 - 0.00241i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 4.84iT - 19T^{2} \)
23 \( 1 + (1.08 - 0.246i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.58 + 0.361i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 3.23iT - 31T^{2} \)
37 \( 1 + (2.10 - 9.23i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (7.61 + 3.66i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.421 - 0.203i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.51 + 1.89i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (6.81 - 1.55i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-7.40 + 3.56i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-14.5 - 3.32i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + (-4.38 + 1.00i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.34 - 3.46i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + (10.4 + 13.1i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (3.92 + 4.92i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 2.91iT - 97T^{2} \)
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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

−11.42086835466664193322954051776, −10.22712515997052665659523474849, −9.753439580982789704540766857699, −8.527274146504520270438733397347, −7.07051421057813576556627943928, −6.46004634621888468480288889293, −5.63307881511744542121458001561, −5.16235643209585241070211234680, −3.27026924747348996413942690052, −2.47761142821073376917615214951, 1.63915497767788662789588539553, 2.58659097596440301287425577817, 3.99978934383725845175700116799, 5.10791219035899597233306369732, 5.67575770723658312317318180371, 6.82543044286158910679937371233, 8.301823395146386287801501492511, 9.692202828716651038413721946472, 10.01754207561275236633983739597, 10.81182821164370039470901815615

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