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Properties

Label 2-21e2-21.20-c3-0-5
Degree $2$
Conductor $441$
Sign $-0.716 - 0.698i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.641i·2-s + 7.58·4-s − 12.0·5-s + 9.99i·8-s − 7.73i·10-s − 43.9i·11-s + 66.9i·13-s + 54.3·16-s − 39.4·17-s + 57.9i·19-s − 91.6·20-s + 28.1·22-s − 46.1i·23-s + 20.7·25-s − 42.9·26-s + ⋯
L(s)  = 1  + 0.226i·2-s + 0.948·4-s − 1.07·5-s + 0.441i·8-s − 0.244i·10-s − 1.20i·11-s + 1.42i·13-s + 0.848·16-s − 0.562·17-s + 0.699i·19-s − 1.02·20-s + 0.272·22-s − 0.418i·23-s + 0.166·25-s − 0.323·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.716 - 0.698i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.716 - 0.698i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.055244068\)
\(L(\frac12)\) \(\approx\) \(1.055244068\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.641iT - 8T^{2} \)
5 \( 1 + 12.0T + 125T^{2} \)
11 \( 1 + 43.9iT - 1.33e3T^{2} \)
13 \( 1 - 66.9iT - 2.19e3T^{2} \)
17 \( 1 + 39.4T + 4.91e3T^{2} \)
19 \( 1 - 57.9iT - 6.85e3T^{2} \)
23 \( 1 + 46.1iT - 1.21e4T^{2} \)
29 \( 1 - 201. iT - 2.43e4T^{2} \)
31 \( 1 - 185. iT - 2.97e4T^{2} \)
37 \( 1 + 410.T + 5.06e4T^{2} \)
41 \( 1 + 408.T + 6.89e4T^{2} \)
43 \( 1 - 129.T + 7.95e4T^{2} \)
47 \( 1 - 514.T + 1.03e5T^{2} \)
53 \( 1 - 437. iT - 1.48e5T^{2} \)
59 \( 1 + 607.T + 2.05e5T^{2} \)
61 \( 1 + 24.5iT - 2.26e5T^{2} \)
67 \( 1 - 441.T + 3.00e5T^{2} \)
71 \( 1 - 1.15e3iT - 3.57e5T^{2} \)
73 \( 1 + 525. iT - 3.89e5T^{2} \)
79 \( 1 + 151.T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + 583.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3iT - 9.12e5T^{2} \)
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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

−11.06352269277613634128238832529, −10.51281613968431472280037335378, −8.900986157517101634100844115627, −8.327904032526377243136374391317, −7.18907754304559251986700975687, −6.62734501027741717520339508635, −5.45201559456708737561417625509, −4.06297099738126028625969132046, −3.09280099162342632993578545078, −1.56329744431272292741306585070, 0.32133736527347235385592524849, 2.04774717491766623636162437233, 3.23126919429808674198279726132, 4.31424261382969497644091519974, 5.59542821177359317332490572044, 6.88782701312635222934610670734, 7.52438600205374410471357480269, 8.298642898584557714526393454611, 9.697836647376901619691392520413, 10.50620595937129366039972992575

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