Properties

Label 2-21e2-1.1-c3-0-47
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.55·2-s + 12.7·4-s − 17.8·5-s + 21.6·8-s − 81.4·10-s + 11.3·11-s − 13.0·13-s − 3.25·16-s − 53.2·17-s − 42.4·19-s − 228.·20-s + 51.9·22-s − 152.·23-s + 194.·25-s − 59.6·26-s − 186.·29-s − 157.·31-s − 188.·32-s − 242.·34-s + 3.74·37-s − 193.·38-s − 387.·40-s + 39.3·41-s + 429.·43-s + 145.·44-s − 692.·46-s − 21.1·47-s + ⋯
L(s)  = 1  + 1.61·2-s + 1.59·4-s − 1.59·5-s + 0.958·8-s − 2.57·10-s + 0.312·11-s − 0.279·13-s − 0.0508·16-s − 0.759·17-s − 0.512·19-s − 2.54·20-s + 0.503·22-s − 1.37·23-s + 1.55·25-s − 0.450·26-s − 1.19·29-s − 0.914·31-s − 1.04·32-s − 1.22·34-s + 0.0166·37-s − 0.825·38-s − 1.53·40-s + 0.149·41-s + 1.52·43-s + 0.498·44-s − 2.22·46-s − 0.0657·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.55T + 8T^{2} \)
5 \( 1 + 17.8T + 125T^{2} \)
11 \( 1 - 11.3T + 1.33e3T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 + 53.2T + 4.91e3T^{2} \)
19 \( 1 + 42.4T + 6.85e3T^{2} \)
23 \( 1 + 152.T + 1.21e4T^{2} \)
29 \( 1 + 186.T + 2.43e4T^{2} \)
31 \( 1 + 157.T + 2.97e4T^{2} \)
37 \( 1 - 3.74T + 5.06e4T^{2} \)
41 \( 1 - 39.3T + 6.89e4T^{2} \)
43 \( 1 - 429.T + 7.95e4T^{2} \)
47 \( 1 + 21.1T + 1.03e5T^{2} \)
53 \( 1 + 365.T + 1.48e5T^{2} \)
59 \( 1 - 226.T + 2.05e5T^{2} \)
61 \( 1 - 651.T + 2.26e5T^{2} \)
67 \( 1 - 145.T + 3.00e5T^{2} \)
71 \( 1 - 368.T + 3.57e5T^{2} \)
73 \( 1 - 608.T + 3.89e5T^{2} \)
79 \( 1 - 910.T + 4.93e5T^{2} \)
83 \( 1 - 327.T + 5.71e5T^{2} \)
89 \( 1 - 37.6T + 7.04e5T^{2} \)
97 \( 1 - 722.T + 9.12e5T^{2} \)
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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

−10.90712981503234240984284358353, −9.299831620851376847187155453988, −8.116394848977815809515306635508, −7.25398054150672174831257204235, −6.34033525614851243078787670340, −5.17975035521971042503835357013, −4.06664294100956133778545499881, −3.77016929711649852100527513874, −2.30782215992531840437107046884, 0, 2.30782215992531840437107046884, 3.77016929711649852100527513874, 4.06664294100956133778545499881, 5.17975035521971042503835357013, 6.34033525614851243078787670340, 7.25398054150672174831257204235, 8.116394848977815809515306635508, 9.299831620851376847187155453988, 10.90712981503234240984284358353

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