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Properties

Label 2-880-11.3-c1-0-7
Degree $2$
Conductor $880$
Sign $-0.939 - 0.343i$
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  + (1.01 + 3.13i)3-s + (0.809 + 0.587i)5-s + (−1.08 + 3.34i)7-s + (−6.34 + 4.61i)9-s + (1.91 + 2.70i)11-s + (3.45 − 2.51i)13-s + (−1.01 + 3.13i)15-s + (−1.28 − 0.934i)17-s + (−1.71 − 5.27i)19-s − 11.5·21-s + 6.39·23-s + (0.309 + 0.951i)25-s + (−12.9 − 9.37i)27-s + (−0.117 + 0.362i)29-s + (−0.615 + 0.446i)31-s + ⋯
L(s)  = 1  + (0.587 + 1.80i)3-s + (0.361 + 0.262i)5-s + (−0.410 + 1.26i)7-s + (−2.11 + 1.53i)9-s + (0.577 + 0.816i)11-s + (0.958 − 0.696i)13-s + (−0.262 + 0.808i)15-s + (−0.311 − 0.226i)17-s + (−0.393 − 1.21i)19-s − 2.52·21-s + 1.33·23-s + (0.0618 + 0.190i)25-s + (−2.48 − 1.80i)27-s + (−0.0218 + 0.0673i)29-s + (−0.110 + 0.0802i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328367 + 1.85325i\)
\(L(\frac12)\) \(\approx\) \(0.328367 + 1.85325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-1.91 - 2.70i)T \)
good3 \( 1 + (-1.01 - 3.13i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.08 - 3.34i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.45 + 2.51i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.28 + 0.934i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.71 + 5.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 + (0.117 - 0.362i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.615 - 0.446i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.448 - 1.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.89 + 5.82i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + (1.33 + 4.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.62 + 4.08i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.92 - 12.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.19 - 5.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + (-5.95 - 4.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.17 + 6.68i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.65 - 4.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.69 + 6.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.451T + 89T^{2} \)
97 \( 1 + (-2.24 + 1.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.41746572037621769775865563473, −9.531127453347271166687969724828, −8.920455489438658199613974206891, −8.632133525682987108524352040578, −7.06168429165983858311183871412, −5.86640939958215537847841280054, −5.15675559866730233402486488106, −4.20944091239351956889653283468, −3.12847978980293193779670340543, −2.43141283534897768680821204705, 0.894878432383776502689485322093, 1.68163082025527649219415883263, 3.14723107886132476039404816423, 4.03328802950578394513163961759, 5.88221784323415626807145619572, 6.49039733799253736833932208943, 7.10675176560281375652539384462, 8.061880529450913168861334394005, 8.713870906164138796936673428060, 9.482699068729691632864833316623

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