Properties

Label 2-8820-21.20-c1-0-41
Degree $2$
Conductor $8820$
Sign $0.0980 + 0.995i$
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 − 2.49i·11-s − 6.80i·13-s + 7.80·17-s + 2.46i·19-s − 3.62i·23-s + 25-s + 2.55i·29-s + 4.25i·31-s + 1.85·37-s − 4.23·41-s + 8.70·43-s + 1.03·47-s + 0.0572i·53-s + 2.49i·55-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.751i·11-s − 1.88i·13-s + 1.89·17-s + 0.564i·19-s − 0.756i·23-s + 0.200·25-s + 0.473i·29-s + 0.764i·31-s + 0.304·37-s − 0.661·41-s + 1.32·43-s + 0.150·47-s + 0.00787i·53-s + 0.336i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8820\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0980 + 0.995i$
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: \((2,\ 8820,\ (\ :1/2),\ 0.0980 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.819521593\)
\(L(\frac12)\) \(\approx\) \(1.819521593\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.49iT - 11T^{2} \)
13 \( 1 + 6.80iT - 13T^{2} \)
17 \( 1 - 7.80T + 17T^{2} \)
19 \( 1 - 2.46iT - 19T^{2} \)
23 \( 1 + 3.62iT - 23T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 - 4.25iT - 31T^{2} \)
37 \( 1 - 1.85T + 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 - 1.03T + 47T^{2} \)
53 \( 1 - 0.0572iT - 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 3.10iT - 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 - 4.15iT - 73T^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 9.96T + 89T^{2} \)
97 \( 1 + 4.91iT - 97T^{2} \)
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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

−7.64076442475977258937443197167, −7.09004062818193450999893815554, −5.99842152760545701950555507200, −5.59510182447658394441249934205, −4.97138369887234859416567200381, −3.84899422547430561457313273067, −3.27755301866129603066248680680, −2.71605361058766414269011815195, −1.25141266647530949161847688497, −0.51988032205579300792582433018, 1.00436066189669112297027053113, 1.93896502656517594827359541844, 2.81040610547617600728806097956, 3.90215854804411241569251055907, 4.20833017461971699762041680272, 5.15631968313107988112350033679, 5.79940381113958548414246129249, 6.74411490545187157672297677918, 7.21419249762441035304703750630, 7.80694708599009706667082401143

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