Properties

Label 2-21e2-1.1-c1-0-10
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  + 2.41·2-s + 3.82·4-s + 0.585·5-s + 4.41·8-s + 1.41·10-s + 2·11-s − 5.41·13-s + 2.99·16-s + 6.24·17-s − 2.82·19-s + 2.24·20-s + 4.82·22-s − 3.65·23-s − 4.65·25-s − 13.0·26-s + 1.17·29-s + 6.82·31-s − 1.58·32-s + 15.0·34-s − 4·37-s − 6.82·38-s + 2.58·40-s − 2.24·41-s − 5.65·43-s + 7.65·44-s − 8.82·46-s + 2.82·47-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.91·4-s + 0.261·5-s + 1.56·8-s + 0.447·10-s + 0.603·11-s − 1.50·13-s + 0.749·16-s + 1.51·17-s − 0.648·19-s + 0.501·20-s + 1.02·22-s − 0.762·23-s − 0.931·25-s − 2.56·26-s + 0.217·29-s + 1.22·31-s − 0.280·32-s + 2.58·34-s − 0.657·37-s − 1.10·38-s + 0.408·40-s − 0.350·41-s − 0.862·43-s + 1.15·44-s − 1.30·46-s + 0.412·47-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

\(L(1)\) \(\approx\) \(3.559174464\)
\(L(\frac12)\) \(\approx\) \(3.559174464\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
5 \( 1 - 0.585T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 - 6.24T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 + 3.75T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 5.89T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 + 5.41T + 97T^{2} \)
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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

−11.67783777386154003194467814591, −10.33521766744250026935737225966, −9.627229635967836355226680600683, −8.088777723242449878848007036447, −7.05205720589200451921747377812, −6.14117276544581238883924624336, −5.27009850611074602684499346558, −4.35394752223632909721139669670, −3.27754524893025953274612190053, −2.07454851217190769150187883915, 2.07454851217190769150187883915, 3.27754524893025953274612190053, 4.35394752223632909721139669670, 5.27009850611074602684499346558, 6.14117276544581238883924624336, 7.05205720589200451921747377812, 8.088777723242449878848007036447, 9.627229635967836355226680600683, 10.33521766744250026935737225966, 11.67783777386154003194467814591

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