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Properties

Label 2-21e2-63.47-c1-0-13
Degree $2$
Conductor $441$
Sign $-0.0920 - 0.995i$
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  + (0.555 − 0.320i)2-s + (0.175 + 1.72i)3-s + (−0.794 + 1.37i)4-s + 2.21·5-s + (0.650 + 0.901i)6-s + 2.30i·8-s + (−2.93 + 0.605i)9-s + (1.22 − 0.709i)10-s + 3.39i·11-s + (−2.50 − 1.12i)12-s + (1.56 − 0.901i)13-s + (0.388 + 3.80i)15-s + (−0.849 − 1.47i)16-s + (−2.98 − 5.16i)17-s + (−1.43 + 1.27i)18-s + (1.42 + 0.822i)19-s + ⋯
L(s)  = 1  + (0.392 − 0.226i)2-s + (0.101 + 0.994i)3-s + (−0.397 + 0.687i)4-s + 0.988·5-s + (0.265 + 0.367i)6-s + 0.813i·8-s + (−0.979 + 0.201i)9-s + (0.388 − 0.224i)10-s + 1.02i·11-s + (−0.724 − 0.325i)12-s + (0.432 − 0.249i)13-s + (0.100 + 0.983i)15-s + (−0.212 − 0.367i)16-s + (−0.723 − 1.25i)17-s + (−0.338 + 0.301i)18-s + (0.326 + 0.188i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0920 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0920 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0920 - 0.995i$
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: \((2,\ 441,\ (\ :1/2),\ -0.0920 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17580 + 1.28955i\)
\(L(\frac12)\) \(\approx\) \(1.17580 + 1.28955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.175 - 1.72i)T \)
7 \( 1 \)
good2 \( 1 + (-0.555 + 0.320i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.21T + 5T^{2} \)
11 \( 1 - 3.39iT - 11T^{2} \)
13 \( 1 + (-1.56 + 0.901i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.98 + 5.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.42 - 0.822i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.37iT - 23T^{2} \)
29 \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.28 - 5.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.849 - 1.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.455 - 0.788i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.96 - 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.123 + 0.213i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.82 + 3.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.39 + 9.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.22 + 0.709i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.99 + 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + (-0.369 + 0.213i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.49 - 4.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.28 + 7.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.26 + 9.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.30 - 3.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.44157644094684532002652927580, −10.29602874845079742437800066619, −9.608121113302203638709273474868, −8.896955337709674663753259237212, −7.890823834914434208443963900751, −6.54616651544273046274687191933, −5.18931033687921196232310945228, −4.68058038039684314808872826537, −3.40720879777106592679685183309, −2.36225856194192754721977439170, 1.04047743631055391684940725280, 2.39160351448945875648413626935, 4.04599174600119302497722916827, 5.54971111862461035533906871153, 6.11576758488133154976854860468, 6.74759772528103245153577840404, 8.287043077069841555832835718950, 8.915743725567573229618675828536, 9.999854687196419027516152702957, 10.85865258396600546518538441028

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