Properties

Label 2-8820-21.20-c1-0-53
Degree $2$
Conductor $8820$
Sign $-0.896 + 0.442i$
Analytic cond. $70.4280$
Root an. cond. $8.39214$
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  − 5-s − 5.98i·11-s − 4.53i·13-s + 5.43·17-s − 5.00i·19-s − 4.86i·23-s + 25-s − 9.24i·29-s − 2.59i·31-s + 9.57·37-s − 4.39·41-s − 8.38·43-s − 5.90·47-s + 2.65i·53-s + 5.98i·55-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80i·11-s − 1.25i·13-s + 1.31·17-s − 1.14i·19-s − 1.01i·23-s + 0.200·25-s − 1.71i·29-s − 0.465i·31-s + 1.57·37-s − 0.686·41-s − 1.27·43-s − 0.860·47-s + 0.365i·53-s + 0.806i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8820\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(70.4280\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8820} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8820,\ (\ :1/2),\ -0.896 + 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.531217683\)
\(L(\frac12)\) \(\approx\) \(1.531217683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 5.98iT - 11T^{2} \)
13 \( 1 + 4.53iT - 13T^{2} \)
17 \( 1 - 5.43T + 17T^{2} \)
19 \( 1 + 5.00iT - 19T^{2} \)
23 \( 1 + 4.86iT - 23T^{2} \)
29 \( 1 + 9.24iT - 29T^{2} \)
31 \( 1 + 2.59iT - 31T^{2} \)
37 \( 1 - 9.57T + 37T^{2} \)
41 \( 1 + 4.39T + 41T^{2} \)
43 \( 1 + 8.38T + 43T^{2} \)
47 \( 1 + 5.90T + 47T^{2} \)
53 \( 1 - 2.65iT - 53T^{2} \)
59 \( 1 - 7.94T + 59T^{2} \)
61 \( 1 + 7.93iT - 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 16.6iT - 73T^{2} \)
79 \( 1 - 4.49T + 79T^{2} \)
83 \( 1 + 0.766T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 7.39iT - 97T^{2} \)
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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

−7.68176879945028982459543480354, −6.71031988517624593756750010565, −6.00521282276500106875243427295, −5.51157257566356159259824664722, −4.69362594102461975117838347477, −3.78252808238238386680550101010, −3.09689537997244932077491518516, −2.56155642303459416916094844174, −0.929658355575968870177124681229, −0.42739144337368547145696264360, 1.45283928296874941009441738819, 1.85214005232197536194080702345, 3.19268031281648382113684297962, 3.75000517856927399090408775523, 4.66661937996632834523368390048, 5.06497211518712651108297375924, 6.06823157539633658415155926528, 6.80741684097281458581802187352, 7.45287041232238549232468894831, 7.79076373470006849164353693527

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