Properties

Label 2-21e2-1.1-c5-0-39
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $70.7292$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.18·2-s + 52.2·4-s − 22.0·5-s + 186.·8-s − 202.·10-s − 416.·11-s + 797.·13-s + 37.3·16-s + 1.37e3·17-s + 2.31e3·19-s − 1.15e3·20-s − 3.82e3·22-s + 955.·23-s − 2.63e3·25-s + 7.32e3·26-s + 7.03e3·29-s + 1.26e3·31-s − 5.61e3·32-s + 1.26e4·34-s + 9.77e3·37-s + 2.12e4·38-s − 4.11e3·40-s + 5.40e3·41-s + 1.96e4·43-s − 2.17e4·44-s + 8.77e3·46-s − 2.05e3·47-s + ⋯
L(s)  = 1  + 1.62·2-s + 1.63·4-s − 0.394·5-s + 1.02·8-s − 0.640·10-s − 1.03·11-s + 1.30·13-s + 0.0364·16-s + 1.15·17-s + 1.46·19-s − 0.645·20-s − 1.68·22-s + 0.376·23-s − 0.844·25-s + 2.12·26-s + 1.55·29-s + 0.235·31-s − 0.970·32-s + 1.87·34-s + 1.17·37-s + 2.38·38-s − 0.406·40-s + 0.501·41-s + 1.62·43-s − 1.69·44-s + 0.611·46-s − 0.135·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.7292\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.682188578\)
\(L(\frac12)\) \(\approx\) \(5.682188578\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.18T + 32T^{2} \)
5 \( 1 + 22.0T + 3.12e3T^{2} \)
11 \( 1 + 416.T + 1.61e5T^{2} \)
13 \( 1 - 797.T + 3.71e5T^{2} \)
17 \( 1 - 1.37e3T + 1.41e6T^{2} \)
19 \( 1 - 2.31e3T + 2.47e6T^{2} \)
23 \( 1 - 955.T + 6.43e6T^{2} \)
29 \( 1 - 7.03e3T + 2.05e7T^{2} \)
31 \( 1 - 1.26e3T + 2.86e7T^{2} \)
37 \( 1 - 9.77e3T + 6.93e7T^{2} \)
41 \( 1 - 5.40e3T + 1.15e8T^{2} \)
43 \( 1 - 1.96e4T + 1.47e8T^{2} \)
47 \( 1 + 2.05e3T + 2.29e8T^{2} \)
53 \( 1 + 1.80e4T + 4.18e8T^{2} \)
59 \( 1 + 7.43e3T + 7.14e8T^{2} \)
61 \( 1 - 3.49e3T + 8.44e8T^{2} \)
67 \( 1 - 1.58e4T + 1.35e9T^{2} \)
71 \( 1 + 5.81e4T + 1.80e9T^{2} \)
73 \( 1 - 3.91e4T + 2.07e9T^{2} \)
79 \( 1 - 9.76e3T + 3.07e9T^{2} \)
83 \( 1 - 7.03e4T + 3.93e9T^{2} \)
89 \( 1 + 1.44e5T + 5.58e9T^{2} \)
97 \( 1 + 7.93e4T + 8.58e9T^{2} \)
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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

−10.71203440210950847007034867516, −9.565896444675696029505016205041, −8.160848140626396505246525551774, −7.40756969004114019803491114645, −6.13951248484428790200474988634, −5.49615048531928508580050510313, −4.49912709503874631425039267451, −3.46765948753357563284501076428, −2.72838301660620829243733551990, −1.01656627101711194647285790961, 1.01656627101711194647285790961, 2.72838301660620829243733551990, 3.46765948753357563284501076428, 4.49912709503874631425039267451, 5.49615048531928508580050510313, 6.13951248484428790200474988634, 7.40756969004114019803491114645, 8.160848140626396505246525551774, 9.565896444675696029505016205041, 10.71203440210950847007034867516

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