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Properties

Label 2-21e2-63.59-c1-0-13
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: $\chi_{441} (374, \cdot )$
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

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

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