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Properties

Label 2-21e2-63.47-c1-0-27
Degree $2$
Conductor $441$
Sign $0.251 + 0.967i$
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  + (1.97 − 1.13i)2-s + (−1.70 + 0.315i)3-s + (1.59 − 2.75i)4-s + 1.43·5-s + (−2.99 + 2.55i)6-s − 2.69i·8-s + (2.80 − 1.07i)9-s + (2.82 − 1.63i)10-s − 3.23i·11-s + (−1.84 + 5.19i)12-s + (4.43 − 2.55i)13-s + (−2.44 + 0.451i)15-s + (0.119 + 0.207i)16-s + (−0.545 − 0.945i)17-s + (4.30 − 5.30i)18-s + (−3.88 − 2.24i)19-s + ⋯
L(s)  = 1  + (1.39 − 0.804i)2-s + (−0.983 + 0.181i)3-s + (0.795 − 1.37i)4-s + 0.641·5-s + (−1.22 + 1.04i)6-s − 0.951i·8-s + (0.933 − 0.357i)9-s + (0.894 − 0.516i)10-s − 0.975i·11-s + (−0.531 + 1.49i)12-s + (1.22 − 0.709i)13-s + (−0.630 + 0.116i)15-s + (0.0298 + 0.0517i)16-s + (−0.132 − 0.229i)17-s + (1.01 − 1.25i)18-s + (−0.891 − 0.514i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.251 + 0.967i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93434 - 1.49529i\)
\(L(\frac12)\) \(\approx\) \(1.93434 - 1.49529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.315i)T \)
7 \( 1 \)
good2 \( 1 + (-1.97 + 1.13i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.43T + 5T^{2} \)
11 \( 1 + 3.23iT - 11T^{2} \)
13 \( 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.545 + 0.945i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.88 + 2.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.00iT - 23T^{2} \)
29 \( 1 + (-1.02 - 0.593i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 - 1.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.119 + 0.207i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.71 - 6.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.11 + 3.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.07 - 3.50i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.73 - 8.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.82 + 1.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.330 - 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 + (-6.33 + 3.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.83 + 3.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.45 - 9.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.84 - 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.07632634836637923654271999204, −10.60727336738921516598793498853, −9.558892024267541140169450017853, −8.222119750122167962394415235756, −6.44939957549835937795190097913, −5.94698350784051624883226549839, −5.14640835812193219628195766436, −4.07731243139047346959059938271, −3.00717990157202107955085909499, −1.35489751295624731059116994899, 1.95403144978647029326690790985, 3.95854215386783067758554222531, 4.65969233300139398438198883891, 5.77441985871631732976062999589, 6.35495406049118210574676965092, 7.01177589204816801089619183205, 8.234438293331555627169791705823, 9.688102317204935669115277233258, 10.61238491761862248050445524860, 11.62755124117991672573635329004

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