Properties

Label 2-21e2-1.1-c1-0-2
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
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 + 5.00·4-s − 7.93·8-s + 5.29·11-s + 11.0·16-s − 14.0·22-s − 5.29·23-s − 5·25-s + 10.5·29-s − 13.2·32-s + 6·37-s + 12·43-s + 26.4·44-s + 14.0·46-s + 13.2·50-s + 10.5·53-s − 28.0·58-s + 13.0·64-s + 4·67-s + 5.29·71-s − 15.8·74-s + 8·79-s − 31.7·86-s − 42.0·88-s − 26.4·92-s − 25.0·100-s − 28.0·106-s + ⋯
L(s)  = 1  − 1.87·2-s + 2.50·4-s − 2.80·8-s + 1.59·11-s + 2.75·16-s − 2.98·22-s − 1.10·23-s − 25-s + 1.96·29-s − 2.33·32-s + 0.986·37-s + 1.82·43-s + 3.98·44-s + 2.06·46-s + 1.87·50-s + 1.45·53-s − 3.67·58-s + 1.62·64-s + 0.488·67-s + 0.627·71-s − 1.84·74-s + 0.900·79-s − 3.42·86-s − 4.47·88-s − 2.75·92-s − 2.50·100-s − 2.71·106-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6475855807\)
\(L(\frac12)\) \(\approx\) \(0.6475855807\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T + 2T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 5.29T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.87499971733367981585631075971, −9.957838963733577647122838300373, −9.350293495462748629510472403158, −8.520751961180749240367235472642, −7.72390490908741287354403524651, −6.70112406592555219308003279582, −6.02340198806629958935966074475, −4.00801398992300897065524229330, −2.38459610907231863518447469134, −1.04262763156758973621639725624, 1.04262763156758973621639725624, 2.38459610907231863518447469134, 4.00801398992300897065524229330, 6.02340198806629958935966074475, 6.70112406592555219308003279582, 7.72390490908741287354403524651, 8.520751961180749240367235472642, 9.350293495462748629510472403158, 9.957838963733577647122838300373, 10.87499971733367981585631075971

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