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Properties

Label 2-21e2-9.7-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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.673 + 1.16i)2-s + (−1.70 − 0.300i)3-s + (0.0923 + 0.160i)4-s + (1.26 + 2.19i)5-s + (1.49 − 1.78i)6-s − 2.94·8-s + (2.81 + 1.02i)9-s − 3.41·10-s + (−0.233 + 0.405i)11-s + (−0.109 − 0.300i)12-s + (2.91 + 5.04i)13-s + (−1.5 − 4.12i)15-s + (1.79 − 3.11i)16-s − 3.87·17-s + (−3.09 + 2.59i)18-s + 2.18·19-s + ⋯
L(s)  = 1  + (−0.476 + 0.825i)2-s + (−0.984 − 0.173i)3-s + (0.0461 + 0.0800i)4-s + (0.566 + 0.980i)5-s + (0.612 − 0.729i)6-s − 1.04·8-s + (0.939 + 0.342i)9-s − 1.07·10-s + (−0.0705 + 0.122i)11-s + (−0.0316 − 0.0868i)12-s + (0.807 + 1.39i)13-s + (−0.387 − 1.06i)15-s + (0.449 − 0.778i)16-s − 0.940·17-s + (−0.729 + 0.612i)18-s + 0.501·19-s + ⋯

Functional equation

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

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: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(-0.706057i\)
\(L(\frac12)\) \(\approx\) \(-0.706057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.300i)T \)
7 \( 1 \)
good2 \( 1 + (0.673 - 1.16i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.233 - 0.405i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.91 - 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (-0.0530 - 0.0918i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.39 - 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (1.11 + 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.66 - 4.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.716T + 53T^{2} \)
59 \( 1 + (-0.368 - 0.637i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.479 + 0.829i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.81 - 8.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.36 - 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 + (6.80 - 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.31315686791459068192442991423, −10.87548770288600803197793536147, −9.655347507703345092580509672444, −8.877538057137307178176062785802, −7.54495543411786010556316356633, −6.72582837895189839214897688800, −6.40382416894103470294944094644, −5.29910962451854833512734116364, −3.73886746560442054309360081203, −2.04411221979611900259145158075, 0.58166333748627541630779119806, 1.77658191995405483126933292561, 3.54394925689353323755237639096, 5.13935601407179292856399639234, 5.67844288519085951184998106691, 6.69637421323626384621846142307, 8.265794452917089945153886080737, 9.198286147817399544931507366877, 9.926362703998840964147268809312, 10.75688007106150788180898748881

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