Properties

Label 2-21e2-1.1-c5-0-10
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  − 8.20·2-s + 35.3·4-s − 29.2·5-s − 27.7·8-s + 239.·10-s − 377.·11-s + 509.·13-s − 904.·16-s − 1.60e3·17-s + 2.15e3·19-s − 1.03e3·20-s + 3.09e3·22-s + 4.43e3·23-s − 2.27e3·25-s − 4.18e3·26-s − 4.77e3·29-s − 7.15e3·31-s + 8.31e3·32-s + 1.31e4·34-s + 5.57e3·37-s − 1.77e4·38-s + 811.·40-s + 9.91e3·41-s − 4.88e3·43-s − 1.33e4·44-s − 3.64e4·46-s − 9.26e3·47-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.10·4-s − 0.522·5-s − 0.153·8-s + 0.758·10-s − 0.940·11-s + 0.836·13-s − 0.883·16-s − 1.34·17-s + 1.37·19-s − 0.578·20-s + 1.36·22-s + 1.74·23-s − 0.726·25-s − 1.21·26-s − 1.05·29-s − 1.33·31-s + 1.43·32-s + 1.95·34-s + 0.669·37-s − 1.98·38-s + 0.0801·40-s + 0.920·41-s − 0.402·43-s − 1.03·44-s − 2.53·46-s − 0.611·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5610354841\)
\(L(\frac12)\) \(\approx\) \(0.5610354841\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.20T + 32T^{2} \)
5 \( 1 + 29.2T + 3.12e3T^{2} \)
11 \( 1 + 377.T + 1.61e5T^{2} \)
13 \( 1 - 509.T + 3.71e5T^{2} \)
17 \( 1 + 1.60e3T + 1.41e6T^{2} \)
19 \( 1 - 2.15e3T + 2.47e6T^{2} \)
23 \( 1 - 4.43e3T + 6.43e6T^{2} \)
29 \( 1 + 4.77e3T + 2.05e7T^{2} \)
31 \( 1 + 7.15e3T + 2.86e7T^{2} \)
37 \( 1 - 5.57e3T + 6.93e7T^{2} \)
41 \( 1 - 9.91e3T + 1.15e8T^{2} \)
43 \( 1 + 4.88e3T + 1.47e8T^{2} \)
47 \( 1 + 9.26e3T + 2.29e8T^{2} \)
53 \( 1 - 9.24e3T + 4.18e8T^{2} \)
59 \( 1 - 1.41e4T + 7.14e8T^{2} \)
61 \( 1 + 1.80e4T + 8.44e8T^{2} \)
67 \( 1 + 4.65e4T + 1.35e9T^{2} \)
71 \( 1 + 6.42e4T + 1.80e9T^{2} \)
73 \( 1 + 6.09e4T + 2.07e9T^{2} \)
79 \( 1 + 7.14e4T + 3.07e9T^{2} \)
83 \( 1 + 1.15e4T + 3.93e9T^{2} \)
89 \( 1 + 7.82e4T + 5.58e9T^{2} \)
97 \( 1 - 1.51e5T + 8.58e9T^{2} \)
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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

−10.22856499981680309066893297429, −9.198472013588200167456640590183, −8.698092993555542857373564258041, −7.58528305537245538532108075881, −7.19619040945382367681675001810, −5.75348354451856441772174188175, −4.48054948826344086378089647401, −3.07588185089115723203885900992, −1.70618293776182515758503806292, −0.48765561596250828757499328245, 0.48765561596250828757499328245, 1.70618293776182515758503806292, 3.07588185089115723203885900992, 4.48054948826344086378089647401, 5.75348354451856441772174188175, 7.19619040945382367681675001810, 7.58528305537245538532108075881, 8.698092993555542857373564258041, 9.198472013588200167456640590183, 10.22856499981680309066893297429

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