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Properties

Label 2-880-44.43-c1-0-10
Degree $2$
Conductor $880$
Sign $0.522 - 0.852i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  + 1.41i·3-s + 5-s + 3.46·7-s + 0.999·9-s + (−1.73 + 2.82i)11-s + 2.44i·13-s + 1.41i·15-s − 7.34i·17-s + 3.46·19-s + 4.89i·21-s + 1.41i·23-s + 25-s + 5.65i·27-s − 4.89i·29-s + (−4.00 − 2.44i)33-s + ⋯
L(s)  = 1  + 0.816i·3-s + 0.447·5-s + 1.30·7-s + 0.333·9-s + (−0.522 + 0.852i)11-s + 0.679i·13-s + 0.365i·15-s − 1.78i·17-s + 0.794·19-s + 1.06i·21-s + 0.294i·23-s + 0.200·25-s + 1.08i·27-s − 0.909i·29-s + (−0.696 − 0.426i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75601 + 0.983773i\)
\(L(\frac12)\) \(\approx\) \(1.75601 + 0.983773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 + (1.73 - 2.82i)T \)
good3 \( 1 - 1.41iT - 3T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 4.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 - 7.07iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 - 9.79iT - 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.02583201393098840342280439571, −9.652934599721379323443129957703, −8.800740238665318029623215049550, −7.59015794087291390089410915166, −7.11720722685696995751387718491, −5.59212409850118962158581672156, −4.81275495260217578093530168311, −4.31989752042345249206070544106, −2.74943591603418109399980349900, −1.53479445132056958053413866036, 1.16574993162747342477669677027, 2.06930311392595200892033454278, 3.49097719524617578774279015979, 4.84567824441719611062601231832, 5.67720864879623676247459389023, 6.53335202427165147795795419822, 7.69834620333828014363464826553, 8.094430455546345563135046881767, 8.931021584470865555179077416761, 10.28961726610643353308854776194

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