Properties

Label 2-21e2-1.1-c3-0-41
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64·2-s − 0.999·4-s − 23.8·8-s + 26.4·11-s − 55.0·16-s + 70·22-s − 216.·23-s − 125·25-s − 264.·29-s + 44.9·32-s − 450·37-s + 180·43-s − 26.4·44-s − 574·46-s − 330.·50-s + 497.·53-s − 700.·58-s + 559·64-s − 740·67-s + 978.·71-s − 1.19e3·74-s − 1.38e3·79-s + 476.·86-s − 630·88-s + 216.·92-s + 124.·100-s + 1.31e3·106-s + ⋯
L(s)  = 1  + 0.935·2-s − 0.124·4-s − 1.05·8-s + 0.725·11-s − 0.859·16-s + 0.678·22-s − 1.96·23-s − 25-s − 1.69·29-s + 0.248·32-s − 1.99·37-s + 0.638·43-s − 0.0906·44-s − 1.83·46-s − 0.935·50-s + 1.28·53-s − 1.58·58-s + 1.09·64-s − 1.34·67-s + 1.63·71-s − 1.87·74-s − 1.97·79-s + 0.597·86-s − 0.763·88-s + 0.245·92-s + 0.124·100-s + 1.20·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.64T + 8T^{2} \)
5 \( 1 + 125T^{2} \)
11 \( 1 - 26.4T + 1.33e3T^{2} \)
13 \( 1 + 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 216.T + 1.21e4T^{2} \)
29 \( 1 + 264.T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 450T + 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 180T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 497.T + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 2.26e5T^{2} \)
67 \( 1 + 740T + 3.00e5T^{2} \)
71 \( 1 - 978.T + 3.57e5T^{2} \)
73 \( 1 + 3.89e5T^{2} \)
79 \( 1 + 1.38e3T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 9.12e5T^{2} \)
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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

−10.22345143672542539377907321659, −9.359889792137238890421481932832, −8.502732674819621743434886646713, −7.33607592955204318734186367298, −6.13706384992762947213800410844, −5.47023633150184299802934924684, −4.18480204734756348458257590005, −3.58098227001465928934103963452, −1.98360564405467700329272522081, 0, 1.98360564405467700329272522081, 3.58098227001465928934103963452, 4.18480204734756348458257590005, 5.47023633150184299802934924684, 6.13706384992762947213800410844, 7.33607592955204318734186367298, 8.502732674819621743434886646713, 9.359889792137238890421481932832, 10.22345143672542539377907321659

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