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Properties

Label 2-21e2-9.7-c1-0-24
Degree $2$
Conductor $441$
Sign $-0.173 + 0.984i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (0.500 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 1.73i·6-s − 3·8-s + (1.5 − 2.59i)9-s + 0.999·10-s + (−2.5 + 4.33i)11-s + (−1.5 − 0.866i)12-s + (−2.5 − 4.33i)13-s + (1.5 + 0.866i)15-s + (0.500 − 0.866i)16-s − 3·17-s + (1.5 + 2.59i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (0.250 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.707i·6-s − 1.06·8-s + (0.5 − 0.866i)9-s + 0.316·10-s + (−0.753 + 1.30i)11-s + (−0.433 − 0.249i)12-s + (−0.693 − 1.20i)13-s + (0.387 + 0.223i)15-s + (0.125 − 0.216i)16-s − 0.727·17-s + (0.353 + 0.612i)18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.173 + 0.984i$
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: \(1\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70371277507289663796721055705, −10.04680456843492734903442569177, −9.015134875689948125772806480985, −7.972592848908292992146851702832, −7.20582544937681542002073887943, −6.24708362972419236706789795104, −5.13918503241466453936853480635, −4.27111317857515609001967569529, −2.65308002061121635953759547720, 0, 1.71499801023174085083386906158, 2.98802852032987997401209236625, 4.74318850001069861386297510355, 5.87729487185656073521172061780, 6.59997082647348295533167961426, 7.57778244473326964945080354131, 8.812356764534351378782297816606, 9.840309209941096572813486996248, 10.79986000589164937841870418537

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