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Properties

Label 2-880-55.32-c1-0-3
Degree $2$
Conductor $880$
Sign $-0.782 + 0.622i$
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  + (−1.70 + 1.70i)3-s + (−0.707 + 2.12i)5-s + (−2.12 + 2.12i)7-s − 2.82i·9-s + (1.41 + 3i)11-s + (3 + 3i)13-s + (−2.41 − 4.82i)15-s + (−5.12 + 5.12i)17-s + 3·19-s − 7.24i·21-s + (0.171 − 0.171i)23-s + (−3.99 − 3i)25-s + (−0.292 − 0.292i)27-s − 1.24·29-s + 7.24·31-s + ⋯
L(s)  = 1  + (−0.985 + 0.985i)3-s + (−0.316 + 0.948i)5-s + (−0.801 + 0.801i)7-s − 0.942i·9-s + (0.426 + 0.904i)11-s + (0.832 + 0.832i)13-s + (−0.623 − 1.24i)15-s + (−1.24 + 1.24i)17-s + 0.688·19-s − 1.58i·21-s + (0.0357 − 0.0357i)23-s + (−0.799 − 0.600i)25-s + (−0.0563 − 0.0563i)27-s − 0.230·29-s + 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224563 - 0.642655i\)
\(L(\frac12)\) \(\approx\) \(0.224563 - 0.642655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 2.12i)T \)
11 \( 1 + (-1.41 - 3i)T \)
good3 \( 1 + (1.70 - 1.70i)T - 3iT^{2} \)
7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (5.12 - 5.12i)T - 17iT^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + (-0.171 + 0.171i)T - 23iT^{2} \)
29 \( 1 + 1.24T + 29T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 + (-0.121 - 0.121i)T + 37iT^{2} \)
41 \( 1 + 1.75iT - 41T^{2} \)
43 \( 1 + (1.24 + 1.24i)T + 43iT^{2} \)
47 \( 1 + (4.41 + 4.41i)T + 47iT^{2} \)
53 \( 1 + (-9.53 + 9.53i)T - 53iT^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 + 7.24iT - 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 + (-6 - 6i)T + 73iT^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + (-7.24 - 7.24i)T + 83iT^{2} \)
89 \( 1 - 5.48iT - 89T^{2} \)
97 \( 1 + (2.24 + 2.24i)T + 97iT^{2} \)
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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

−10.61659401905975393074637471498, −9.947280113968080212954624165080, −9.256296262201675908449837566577, −8.258643583995313623842801032673, −6.67834251637363705626034991797, −6.51313726464294768072955501127, −5.46276177116541330898254779721, −4.28743279625785067767227066090, −3.65054225635300409941449345482, −2.20441190914380371165426490978, 0.44244744830081526000672590208, 1.12178366515243437860651912448, 3.12630137295924468585481008867, 4.29886924963110064610778634530, 5.41902109677116772495506612713, 6.21005093271034612467327656325, 6.93752094915858684748858485983, 7.78657064392480864458044005699, 8.733417342154764485099685404902, 9.576205843815415011659229353503

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