Properties

Label 2-21e2-63.58-c1-0-15
Degree $2$
Conductor $441$
Sign $0.467 + 0.883i$
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  + 0.670·2-s + (−1.65 − 0.518i)3-s − 1.55·4-s + (0.712 + 1.23i)5-s + (−1.10 − 0.347i)6-s − 2.38·8-s + (2.46 + 1.71i)9-s + (0.477 + 0.827i)10-s + (2.46 − 4.27i)11-s + (2.56 + 0.803i)12-s + (1.37 − 2.38i)13-s + (−0.537 − 2.40i)15-s + 1.50·16-s + (−0.559 − 0.969i)17-s + (1.65 + 1.14i)18-s + (2.00 − 3.47i)19-s + ⋯
L(s)  = 1  + 0.473·2-s + (−0.954 − 0.299i)3-s − 0.775·4-s + (0.318 + 0.551i)5-s + (−0.452 − 0.141i)6-s − 0.841·8-s + (0.820 + 0.571i)9-s + (0.151 + 0.261i)10-s + (0.743 − 1.28i)11-s + (0.739 + 0.232i)12-s + (0.381 − 0.661i)13-s + (−0.138 − 0.621i)15-s + 0.376·16-s + (−0.135 − 0.235i)17-s + (0.389 + 0.270i)18-s + (0.460 − 0.797i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.467 + 0.883i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.467 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899510 - 0.541532i\)
\(L(\frac12)\) \(\approx\) \(0.899510 - 0.541532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 + 0.518i)T \)
7 \( 1 \)
good2 \( 1 - 0.670T + 2T^{2} \)
5 \( 1 + (-0.712 - 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.46 + 4.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.37 + 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.559 + 0.969i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.71 + 4.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.47T + 47T^{2} \)
53 \( 1 + (0.410 + 0.710i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.58T + 59T^{2} \)
61 \( 1 + 0.0752T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (5.34 + 9.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.76 - 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.70 + 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96806936577464602436511724380, −10.35327710427352359909944479805, −9.153419427412355583109827082807, −8.296813022125906078756320569462, −6.89387003609196312490772073408, −6.08802450572581661957389865480, −5.35961297480120390331645677837, −4.22384326682770742445562217467, −2.97486068629253709505070729674, −0.74649135379333170155553149136, 1.45163335863793393107762228213, 3.86210688314794208280573218652, 4.47955894545793981546059030713, 5.46572822167765134514227643897, 6.23950693287166173732304833621, 7.43470180845140759889298142979, 8.883774941387223111795423869561, 9.572276764470866165823496342756, 10.19201683038031346012996151468, 11.62717926692534843738177479820

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