Properties

Label 2-8820-21.20-c1-0-1
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 + 0.286i·11-s + 6.53i·13-s − 1.74·17-s + 2.17i·19-s − 2.66i·23-s + 25-s + 4.02i·29-s + 8.98i·31-s − 3.68·37-s − 9.19·41-s − 6.70·43-s + 11.4·47-s − 5.29i·53-s − 0.286i·55-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.0863i·11-s + 1.81i·13-s − 0.422·17-s + 0.499i·19-s − 0.555i·23-s + 0.200·25-s + 0.746i·29-s + 1.61i·31-s − 0.606·37-s − 1.43·41-s − 1.02·43-s + 1.66·47-s − 0.727i·53-s − 0.0385i·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

\(L(1)\) \(\approx\) \(0.2032540867\)
\(L(\frac12)\) \(\approx\) \(0.2032540867\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.286iT - 11T^{2} \)
13 \( 1 - 6.53iT - 13T^{2} \)
17 \( 1 + 1.74T + 17T^{2} \)
19 \( 1 - 2.17iT - 19T^{2} \)
23 \( 1 + 2.66iT - 23T^{2} \)
29 \( 1 - 4.02iT - 29T^{2} \)
31 \( 1 - 8.98iT - 31T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
41 \( 1 + 9.19T + 41T^{2} \)
43 \( 1 + 6.70T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 5.29iT - 53T^{2} \)
59 \( 1 + 2.20T + 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 + 0.709T + 67T^{2} \)
71 \( 1 + 7.64iT - 71T^{2} \)
73 \( 1 + 4.28iT - 73T^{2} \)
79 \( 1 + 1.00T + 79T^{2} \)
83 \( 1 - 4.03T + 83T^{2} \)
89 \( 1 - 3.28T + 89T^{2} \)
97 \( 1 + 1.29iT - 97T^{2} \)
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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

−8.219832415796963501108872561482, −7.34259837802287889686579236425, −6.71319965932809969819693525842, −6.39989520700984410625216541980, −5.15279031187546315843864286236, −4.73810075853085891554059916904, −3.87045683083847441842578698270, −3.28734550413966289943079576782, −2.12220333950259182050293188751, −1.45585146461531174200398765782, 0.05198036614835496817113155921, 0.978150336955341883212302911253, 2.27298836006486823725495668523, 3.02560083681081539247416945911, 3.76922042936352670722296908407, 4.52434106813245145989401525633, 5.40620839036223090689292111335, 5.84198504498144598970058980093, 6.75930270842137803298053880931, 7.48834148349950863978057856386

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