Properties

Label 2-21e2-1.1-c5-0-52
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.12·2-s + 34.0·4-s + 17.4·5-s − 16.5·8-s − 141.·10-s + 114.·11-s + 205.·13-s − 954.·16-s − 757.·17-s + 1.01e3·19-s + 592.·20-s − 928.·22-s + 916.·23-s − 2.82e3·25-s − 1.66e3·26-s + 1.09e3·29-s − 8.23e3·31-s + 8.28e3·32-s + 6.15e3·34-s − 1.07e4·37-s − 8.23e3·38-s − 287.·40-s + 1.87e4·41-s − 4.64e3·43-s + 3.88e3·44-s − 7.44e3·46-s − 1.39e4·47-s + ⋯
L(s)  = 1  − 1.43·2-s + 1.06·4-s + 0.311·5-s − 0.0912·8-s − 0.447·10-s + 0.284·11-s + 0.336·13-s − 0.932·16-s − 0.635·17-s + 0.644·19-s + 0.331·20-s − 0.409·22-s + 0.361·23-s − 0.902·25-s − 0.483·26-s + 0.241·29-s − 1.53·31-s + 1.43·32-s + 0.912·34-s − 1.28·37-s − 0.925·38-s − 0.0284·40-s + 1.74·41-s − 0.382·43-s + 0.302·44-s − 0.518·46-s − 0.922·47-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: \(1\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.12T + 32T^{2} \)
5 \( 1 - 17.4T + 3.12e3T^{2} \)
11 \( 1 - 114.T + 1.61e5T^{2} \)
13 \( 1 - 205.T + 3.71e5T^{2} \)
17 \( 1 + 757.T + 1.41e6T^{2} \)
19 \( 1 - 1.01e3T + 2.47e6T^{2} \)
23 \( 1 - 916.T + 6.43e6T^{2} \)
29 \( 1 - 1.09e3T + 2.05e7T^{2} \)
31 \( 1 + 8.23e3T + 2.86e7T^{2} \)
37 \( 1 + 1.07e4T + 6.93e7T^{2} \)
41 \( 1 - 1.87e4T + 1.15e8T^{2} \)
43 \( 1 + 4.64e3T + 1.47e8T^{2} \)
47 \( 1 + 1.39e4T + 2.29e8T^{2} \)
53 \( 1 - 2.93e4T + 4.18e8T^{2} \)
59 \( 1 + 3.03e4T + 7.14e8T^{2} \)
61 \( 1 + 1.86e4T + 8.44e8T^{2} \)
67 \( 1 - 1.99e4T + 1.35e9T^{2} \)
71 \( 1 - 5.73e4T + 1.80e9T^{2} \)
73 \( 1 - 6.01e4T + 2.07e9T^{2} \)
79 \( 1 - 3.57e4T + 3.07e9T^{2} \)
83 \( 1 - 8.66e4T + 3.93e9T^{2} \)
89 \( 1 + 4.29e4T + 5.58e9T^{2} \)
97 \( 1 + 2.06e4T + 8.58e9T^{2} \)
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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

−9.577388713198222770160435735172, −9.148878605908269900699617995335, −8.193413070485824763929049720463, −7.32456497746855877960317003148, −6.43717613634165881415478287496, −5.20065512487099792416893349620, −3.79877785017424152885525445251, −2.23976325241379451568608829320, −1.23180965507702294376864588330, 0, 1.23180965507702294376864588330, 2.23976325241379451568608829320, 3.79877785017424152885525445251, 5.20065512487099792416893349620, 6.43717613634165881415478287496, 7.32456497746855877960317003148, 8.193413070485824763929049720463, 9.148878605908269900699617995335, 9.577388713198222770160435735172

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