Properties

Label 2-21e2-63.4-c1-0-29
Degree $2$
Conductor $441$
Sign $-0.0325 + 0.999i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0325 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0325 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0325 + 0.999i$
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: \((2,\ 441,\ (\ :1/2),\ -0.0325 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43439 - 1.48188i\)
\(L(\frac12)\) \(\approx\) \(1.43439 - 1.48188i\)
\(L(\frac{3}{2})\) 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

−11.23806576730002292674023602935, −10.24013077568616745870123735855, −8.858278874482561887528569176842, −8.209858634475618131336337701233, −7.05341780745715430789923810993, −6.48732501739678464433656291531, −4.92204200857210029562907939122, −3.58929104641524734217677124779, −2.66693953124041462758002435508, −1.34118799244652637934992588969, 1.97437562527834566999589728948, 3.91543598529055002234176551908, 4.40019129210135170238558074242, 5.91426152007024201786279460188, 6.24960740176496462747553837295, 7.74511391503461439724437759708, 8.531014922032146649629389794168, 9.775514781736352577563097724993, 10.20738206499657641897633555327, 11.33018962807507029378772943654

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