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Properties

Label 2-21e2-21.20-c1-0-11
Degree $2$
Conductor $441$
Sign $-0.970 + 0.239i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.44·5-s − 2.82i·8-s + 3.46i·10-s − 1.41i·11-s − 5.19i·13-s − 4.00·16-s − 4.89·17-s + 1.73i·19-s − 2.00·22-s − 5.65i·23-s + 0.999·25-s − 7.34·26-s + 2.82i·29-s − 1.73i·31-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.09·5-s − 0.999i·8-s + 1.09i·10-s − 0.426i·11-s − 1.44i·13-s − 1.00·16-s − 1.18·17-s + 0.397i·19-s − 0.426·22-s − 1.17i·23-s + 0.199·25-s − 1.44·26-s + 0.525i·29-s − 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.970 + 0.239i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117865 - 0.970985i\)
\(L(\frac12)\) \(\approx\) \(0.117865 - 0.970985i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41iT - 2T^{2} \)
5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 - 4.89T + 89T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−10.87710575060451513406946005065, −10.20456355763436552593865017412, −8.946459551582198045024373078681, −8.032555063951711344137063250386, −7.12166478652107226530339109384, −5.96535343522107660029560353359, −4.44389664720983271010962041782, −3.51952462200294649916587556444, −2.48901394909076953594804391716, −0.59536133950747632232977552167, 2.20586491300823955334880921721, 3.95544712116567946792405562081, 4.82912548424657138651501699310, 6.15771216219854371217483104082, 7.06990121167278531388781701873, 7.58365130693717008466942491144, 8.637050573571244696593774065092, 9.396728147619794379880482082564, 10.93261386334468410026006189788, 11.50612199599834516613784709871

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