Properties

Label 2-21e2-21.20-c3-0-18
Degree $2$
Conductor $441$
Sign $-0.0980 - 0.995i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.82i·2-s + 4.68·4-s + 15.0·5-s + 23.0i·8-s + 27.4i·10-s − 9.89i·11-s + 67.8i·13-s − 4.54·16-s − 70.1·17-s + 61.4i·19-s + 70.7·20-s + 18.0·22-s + 131. i·23-s + 102.·25-s − 123.·26-s + ⋯
L(s)  = 1  + 0.643i·2-s + 0.585·4-s + 1.34·5-s + 1.02i·8-s + 0.868i·10-s − 0.271i·11-s + 1.44i·13-s − 0.0710·16-s − 1.00·17-s + 0.742i·19-s + 0.790·20-s + 0.174·22-s + 1.19i·23-s + 0.821·25-s − 0.932·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0980 - 0.995i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.0980 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.887808788\)
\(L(\frac12)\) \(\approx\) \(2.887808788\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.82iT - 8T^{2} \)
5 \( 1 - 15.0T + 125T^{2} \)
11 \( 1 + 9.89iT - 1.33e3T^{2} \)
13 \( 1 - 67.8iT - 2.19e3T^{2} \)
17 \( 1 + 70.1T + 4.91e3T^{2} \)
19 \( 1 - 61.4iT - 6.85e3T^{2} \)
23 \( 1 - 131. iT - 1.21e4T^{2} \)
29 \( 1 + 158. iT - 2.43e4T^{2} \)
31 \( 1 + 76.4iT - 2.97e4T^{2} \)
37 \( 1 - 348.T + 5.06e4T^{2} \)
41 \( 1 - 138.T + 6.89e4T^{2} \)
43 \( 1 - 539.T + 7.95e4T^{2} \)
47 \( 1 + 223.T + 1.03e5T^{2} \)
53 \( 1 + 530. iT - 1.48e5T^{2} \)
59 \( 1 + 542.T + 2.05e5T^{2} \)
61 \( 1 - 134. iT - 2.26e5T^{2} \)
67 \( 1 - 320.T + 3.00e5T^{2} \)
71 \( 1 - 416. iT - 3.57e5T^{2} \)
73 \( 1 - 545. iT - 3.89e5T^{2} \)
79 \( 1 + 322.T + 4.93e5T^{2} \)
83 \( 1 - 885.T + 5.71e5T^{2} \)
89 \( 1 + 1.62e3T + 7.04e5T^{2} \)
97 \( 1 + 739. iT - 9.12e5T^{2} \)
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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

−11.09083334166926471687574339901, −9.815918210410845524649503115187, −9.241541962329063674808179931065, −8.094917647994419294438118751729, −7.04068886314559191958666083226, −6.17257876049388053959134137337, −5.68203637002544184191914456962, −4.28095923822032074205606443372, −2.49058399312089437381333562781, −1.67379280233725219534066308983, 0.916128477039177770364381563902, 2.25113243780919595711980064934, 2.93686965112647913399963872161, 4.59790152067986186448067704217, 5.84498142623530173229770753424, 6.55043910559509742619228426041, 7.62673642274528770675388703216, 8.977540225139489844484517002377, 9.738926324434174157351830614884, 10.70321675767986565440813032407

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