Properties

Label 2-21e2-7.4-c1-0-4
Degree $2$
Conductor $441$
Sign $-0.605 - 0.795i$
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 + 1.73i)2-s + (−0.999 + 1.73i)4-s + (1 + 1.73i)5-s + (−1.99 + 3.46i)10-s + (−1 + 1.73i)11-s − 13-s + (1.99 + 3.46i)16-s + (0.5 + 0.866i)19-s − 3.99·20-s − 3.99·22-s + (0.500 − 0.866i)25-s + (−1 − 1.73i)26-s − 4·29-s + (4.5 − 7.79i)31-s + (−3.99 + 6.92i)32-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.447 + 0.774i)5-s + (−0.632 + 1.09i)10-s + (−0.301 + 0.522i)11-s − 0.277·13-s + (0.499 + 0.866i)16-s + (0.114 + 0.198i)19-s − 0.894·20-s − 0.852·22-s + (0.100 − 0.173i)25-s + (−0.196 − 0.339i)26-s − 0.742·29-s + (0.808 − 1.39i)31-s + (−0.707 + 1.22i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.947827 + 1.91199i\)
\(L(\frac12)\) \(\approx\) \(0.947827 + 1.91199i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (8 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−11.47427252918429605360581773829, −10.42037856594733381649778852369, −9.722738693879243613827129138377, −8.345616257938071359097484650533, −7.43218123963546323519361995834, −6.72508313428073797690881419585, −5.87295572801602683615539219319, −4.96952953562237227176871892073, −3.81829038348525041566569393420, −2.27689263339679432504867999504, 1.23016460075341789433949643971, 2.56147766452243104873584991151, 3.70354457972616565365145955591, 4.90003992433324134215765779852, 5.52760332090179227702685584001, 7.01842397817773169064687649106, 8.300831912792073410911994983002, 9.233153361206338198543210490442, 10.19108765328872676940965825392, 10.88381478687683212611247892420

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