Properties

Label 2-21e2-1.1-c5-0-11
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  + 2.64·2-s − 25·4-s − 150.·8-s − 799.·11-s + 401·16-s − 2.11e3·22-s − 1.10e3·23-s − 3.12e3·25-s + 5.38e3·29-s + 5.88e3·32-s + 8.88e3·37-s − 1.17e4·43-s + 1.99e4·44-s − 2.92e3·46-s − 8.26e3·50-s − 3.27e4·53-s + 1.42e4·58-s + 2.74e3·64-s + 6.93e4·67-s + 8.49e4·71-s + 2.35e4·74-s + 8.01e4·79-s − 3.10e4·86-s + 1.20e5·88-s + 2.76e4·92-s + 7.81e4·100-s − 8.65e4·106-s + ⋯
L(s)  = 1  + 0.467·2-s − 0.781·4-s − 0.833·8-s − 1.99·11-s + 0.391·16-s − 0.931·22-s − 0.435·23-s − 25-s + 1.18·29-s + 1.01·32-s + 1.06·37-s − 0.968·43-s + 1.55·44-s − 0.203·46-s − 0.467·50-s − 1.59·53-s + 0.556·58-s + 0.0837·64-s + 1.88·67-s + 1.99·71-s + 0.499·74-s + 1.44·79-s − 0.453·86-s + 1.65·88-s + 0.340·92-s + 0.781·100-s − 0.748·106-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

\(L(3)\) \(\approx\) \(1.184525928\)
\(L(\frac12)\) \(\approx\) \(1.184525928\)
\(L(\frac{7}{2})\) 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 \( 1 - 2.64T + 32T^{2} \)
5 \( 1 + 3.12e3T^{2} \)
11 \( 1 + 799.T + 1.61e5T^{2} \)
13 \( 1 + 3.71e5T^{2} \)
17 \( 1 + 1.41e6T^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 + 1.10e3T + 6.43e6T^{2} \)
29 \( 1 - 5.38e3T + 2.05e7T^{2} \)
31 \( 1 + 2.86e7T^{2} \)
37 \( 1 - 8.88e3T + 6.93e7T^{2} \)
41 \( 1 + 1.15e8T^{2} \)
43 \( 1 + 1.17e4T + 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 + 3.27e4T + 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 8.44e8T^{2} \)
67 \( 1 - 6.93e4T + 1.35e9T^{2} \)
71 \( 1 - 8.49e4T + 1.80e9T^{2} \)
73 \( 1 + 2.07e9T^{2} \)
79 \( 1 - 8.01e4T + 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 + 8.58e9T^{2} \)
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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

−10.19291046148522459549216267902, −9.576159980438837808377951025933, −8.281773270098613762436877864159, −7.84704461165404591598995568846, −6.33687991319700138749495097124, −5.34497644825417837060113296342, −4.66678012638136598455842485655, −3.44330473172198677993418964807, −2.36312681513080519248858569483, −0.50584985409870227403924018514, 0.50584985409870227403924018514, 2.36312681513080519248858569483, 3.44330473172198677993418964807, 4.66678012638136598455842485655, 5.34497644825417837060113296342, 6.33687991319700138749495097124, 7.84704461165404591598995568846, 8.281773270098613762436877864159, 9.576159980438837808377951025933, 10.19291046148522459549216267902

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