Properties

Label 2-21e2-63.4-c1-0-10
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  + (0.551 − 0.955i)2-s + (−0.454 + 1.67i)3-s + (0.391 + 0.678i)4-s + 0.105·5-s + (1.34 + 1.35i)6-s + 3.07·8-s + (−2.58 − 1.52i)9-s + (0.0581 − 0.100i)10-s + 3.33·11-s + (−1.31 + 0.346i)12-s + (−1.23 + 2.14i)13-s + (−0.0479 + 0.176i)15-s + (0.909 − 1.57i)16-s + (−0.806 + 1.39i)17-s + (−2.87 + 1.63i)18-s + (3.84 + 6.65i)19-s + ⋯
L(s)  = 1  + (0.389 − 0.675i)2-s + (−0.262 + 0.964i)3-s + (0.195 + 0.339i)4-s + 0.0471·5-s + (0.549 + 0.553i)6-s + 1.08·8-s + (−0.862 − 0.506i)9-s + (0.0183 − 0.0318i)10-s + 1.00·11-s + (−0.378 + 0.0999i)12-s + (−0.343 + 0.595i)13-s + (−0.0123 + 0.0455i)15-s + (0.227 − 0.393i)16-s + (−0.195 + 0.338i)17-s + (−0.678 + 0.384i)18-s + (0.881 + 1.52i)19-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: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62607 + 0.652336i\)
\(L(\frac12)\) \(\approx\) \(1.62607 + 0.652336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.454 - 1.67i)T \)
7 \( 1 \)
good2 \( 1 + (-0.551 + 0.955i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.105T + 5T^{2} \)
11 \( 1 - 3.33T + 11T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.806 - 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.84 - 6.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 + (-4.64 - 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.63 + 8.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.991 - 1.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.59 + 2.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.98 + 8.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.22 + 3.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + (-2.36 + 4.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.84 - 6.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.584 + 1.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.90 + 3.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.42459867716367684039929595758, −10.40047630640857172419662122605, −9.719079564185890262946036827107, −8.726215455181260736690689778456, −7.58913284277143965077381706766, −6.39198569845942064892655168584, −5.24861443227336768146993969951, −4.00622879120156516669256768813, −3.56620717172160797329782808640, −1.92509546428348752165722423080, 1.13044188569266962923093402170, 2.64613469682347014554576030212, 4.51536339064625332274750368108, 5.55687204050824294658993949219, 6.35740710368187865110137338146, 7.14532477649597675840696425124, 7.84526721243045754174993399200, 9.098938912593347643135852026780, 10.16145950893284269155816836762, 11.31392862222843299743533561847

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