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Properties

Label 2-880-11.3-c1-0-0
Degree $2$
Conductor $880$
Sign $0.0154 - 0.999i$
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  + (−0.768 − 2.36i)3-s + (−0.809 − 0.587i)5-s + (0.133 − 0.410i)7-s + (−2.57 + 1.87i)9-s + (−1.92 + 2.69i)11-s + (−3.19 + 2.31i)13-s + (−0.768 + 2.36i)15-s + (−2.76 − 2.01i)17-s + (1.89 + 5.82i)19-s − 1.07·21-s − 4.17·23-s + (0.309 + 0.951i)25-s + (0.374 + 0.272i)27-s + (−0.195 + 0.602i)29-s + (8.34 − 6.06i)31-s + ⋯
L(s)  = 1  + (−0.443 − 1.36i)3-s + (−0.361 − 0.262i)5-s + (0.0504 − 0.155i)7-s + (−0.859 + 0.624i)9-s + (−0.581 + 0.813i)11-s + (−0.884 + 0.642i)13-s + (−0.198 + 0.610i)15-s + (−0.671 − 0.487i)17-s + (0.434 + 1.33i)19-s − 0.234·21-s − 0.869·23-s + (0.0618 + 0.190i)25-s + (0.0720 + 0.0523i)27-s + (−0.0363 + 0.111i)29-s + (1.49 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0154 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0154 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.0154 - 0.999i$
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: \((2,\ 880,\ (\ :1/2),\ 0.0154 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152103 + 0.149772i\)
\(L(\frac12)\) \(\approx\) \(0.152103 + 0.149772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 - 2.69i)T \)
good3 \( 1 + (0.768 + 2.36i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.133 + 0.410i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.19 - 2.31i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.76 + 2.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.89 - 5.82i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.17T + 23T^{2} \)
29 \( 1 + (0.195 - 0.602i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.34 + 6.06i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.26 - 6.98i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.821 - 2.52i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (0.917 + 2.82i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.24 + 3.08i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.35 - 10.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.51 + 6.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + (8.29 + 6.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.44 - 10.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.02 - 4.37i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.37 - 4.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.34T + 89T^{2} \)
97 \( 1 + (-1.41 + 1.03i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.25742321912406297456575522857, −9.646422495725339175084111981665, −8.284947731156168041148514408749, −7.71495679496888457488555855068, −6.99807240507471531461967019156, −6.24293642615121304675761056915, −5.12411425909023672093874287364, −4.20234704408636821632476887829, −2.52729418767971309227522581782, −1.50938141706939412122704979143, 0.10620060514709802533262235074, 2.64120677424611524689982570053, 3.59812504780125606295845704692, 4.69052643286961136999075110268, 5.26549957954165031597256642304, 6.29136094507399731739495486279, 7.42578758464134661249766433314, 8.426443407092067946393799774442, 9.182815628640121048496777101449, 10.25627975204362924311164726604

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