Properties

Label 2-21e2-1.1-c3-0-22
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  − 4.53·2-s + 12.5·4-s − 13.4·5-s − 20.5·8-s + 61.0·10-s − 0.813·11-s − 34.9·13-s − 7.21·16-s + 117.·17-s + 93.2·19-s − 168.·20-s + 3.68·22-s − 120.·23-s + 56.6·25-s + 158.·26-s − 8.56·29-s + 82.1·31-s + 196.·32-s − 533.·34-s + 28.8·37-s − 422.·38-s + 276.·40-s − 70.5·41-s + 417.·43-s − 10.1·44-s + 544.·46-s + 338.·47-s + ⋯
L(s)  = 1  − 1.60·2-s + 1.56·4-s − 1.20·5-s − 0.907·8-s + 1.93·10-s − 0.0222·11-s − 0.745·13-s − 0.112·16-s + 1.67·17-s + 1.12·19-s − 1.88·20-s + 0.0357·22-s − 1.09·23-s + 0.453·25-s + 1.19·26-s − 0.0548·29-s + 0.475·31-s + 1.08·32-s − 2.69·34-s + 0.128·37-s − 1.80·38-s + 1.09·40-s − 0.268·41-s + 1.47·43-s − 0.0349·44-s + 1.74·46-s + 1.04·47-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 \( 1 + 4.53T + 8T^{2} \)
5 \( 1 + 13.4T + 125T^{2} \)
11 \( 1 + 0.813T + 1.33e3T^{2} \)
13 \( 1 + 34.9T + 2.19e3T^{2} \)
17 \( 1 - 117.T + 4.91e3T^{2} \)
19 \( 1 - 93.2T + 6.85e3T^{2} \)
23 \( 1 + 120.T + 1.21e4T^{2} \)
29 \( 1 + 8.56T + 2.43e4T^{2} \)
31 \( 1 - 82.1T + 2.97e4T^{2} \)
37 \( 1 - 28.8T + 5.06e4T^{2} \)
41 \( 1 + 70.5T + 6.89e4T^{2} \)
43 \( 1 - 417.T + 7.95e4T^{2} \)
47 \( 1 - 338.T + 1.03e5T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 + 94.1T + 2.05e5T^{2} \)
61 \( 1 - 120.T + 2.26e5T^{2} \)
67 \( 1 + 792.T + 3.00e5T^{2} \)
71 \( 1 + 449.T + 3.57e5T^{2} \)
73 \( 1 - 469.T + 3.89e5T^{2} \)
79 \( 1 + 1.01e3T + 4.93e5T^{2} \)
83 \( 1 + 104.T + 5.71e5T^{2} \)
89 \( 1 + 1.57e3T + 7.04e5T^{2} \)
97 \( 1 + 550.T + 9.12e5T^{2} \)
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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

−10.03162711183201112058146784909, −9.442773275033597273530809292592, −8.270447979425849665629482254191, −7.69413932940481052176974085823, −7.17605772906878351180340814396, −5.66839635145663329028065104035, −4.18298112535309774964586306432, −2.85327004240489439611812802860, −1.18827267254064626593880853132, 0, 1.18827267254064626593880853132, 2.85327004240489439611812802860, 4.18298112535309774964586306432, 5.66839635145663329028065104035, 7.17605772906878351180340814396, 7.69413932940481052176974085823, 8.270447979425849665629482254191, 9.442773275033597273530809292592, 10.03162711183201112058146784909

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