Properties

Label 2-8820-21.20-c1-0-6
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 − 2.23i·11-s − 4.92i·13-s − 3.83·17-s + 5.96i·19-s − 4.24i·23-s + 25-s + 6.73i·29-s − 1.80i·31-s − 4.32·37-s − 2.50·41-s + 4.96·43-s − 10.8·47-s + 4.50i·53-s + 2.23i·55-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.673i·11-s − 1.36i·13-s − 0.931·17-s + 1.36i·19-s − 0.886i·23-s + 0.200·25-s + 1.25i·29-s − 0.324i·31-s − 0.711·37-s − 0.391·41-s + 0.757·43-s − 1.58·47-s + 0.618i·53-s + 0.301i·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

\(L(1)\) \(\approx\) \(0.8146164486\)
\(L(\frac12)\) \(\approx\) \(0.8146164486\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + 4.92iT - 13T^{2} \)
17 \( 1 + 3.83T + 17T^{2} \)
19 \( 1 - 5.96iT - 19T^{2} \)
23 \( 1 + 4.24iT - 23T^{2} \)
29 \( 1 - 6.73iT - 29T^{2} \)
31 \( 1 + 1.80iT - 31T^{2} \)
37 \( 1 + 4.32T + 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 - 4.96T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 4.50iT - 53T^{2} \)
59 \( 1 - 3.99T + 59T^{2} \)
61 \( 1 - 6.25iT - 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 8.87iT - 71T^{2} \)
73 \( 1 + 5.01iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 0.295T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 - 1.05iT - 97T^{2} \)
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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

−7.943384777021780845260640505206, −7.34790581996585727777802814810, −6.46930758689394992375844561152, −5.90243536819656259315617900944, −5.15642201065999892382632210754, −4.42320202801249861284536046854, −3.50113667942524204014597495589, −3.05250862881368103417150294832, −1.95521245572495414740297147122, −0.834997819906089008323018067760, 0.22771734098233641415276551736, 1.64658022955209335018379354247, 2.33739575669580314889070589710, 3.32195334965170295803890395109, 4.26572753598472900252478396844, 4.60090027878865132642119534129, 5.42664722889634938758646474398, 6.50611103124636800516356548296, 6.86759608692490998804201193547, 7.46688272037342276690234124681

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