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Properties

Label 2-880-11.4-c1-0-3
Degree $2$
Conductor $880$
Sign $-0.202 - 0.979i$
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  + (−0.453 + 1.39i)3-s + (−0.809 + 0.587i)5-s + (−1.39 − 4.30i)7-s + (0.686 + 0.498i)9-s + (2.39 + 2.29i)11-s + (0.924 + 0.671i)13-s + (−0.453 − 1.39i)15-s + (−2.72 + 1.98i)17-s + (−1.88 + 5.78i)19-s + 6.63·21-s + 5.45·23-s + (0.309 − 0.951i)25-s + (−4.56 + 3.31i)27-s + (1.02 + 3.15i)29-s + (1.44 + 1.05i)31-s + ⋯
L(s)  = 1  + (−0.261 + 0.805i)3-s + (−0.361 + 0.262i)5-s + (−0.528 − 1.62i)7-s + (0.228 + 0.166i)9-s + (0.723 + 0.690i)11-s + (0.256 + 0.186i)13-s + (−0.117 − 0.360i)15-s + (−0.661 + 0.480i)17-s + (−0.431 + 1.32i)19-s + 1.44·21-s + 1.13·23-s + (0.0618 − 0.190i)25-s + (−0.878 + 0.638i)27-s + (0.190 + 0.586i)29-s + (0.260 + 0.189i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709280 + 0.870760i\)
\(L(\frac12)\) \(\approx\) \(0.709280 + 0.870760i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 - 2.29i)T \)
good3 \( 1 + (0.453 - 1.39i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.39 + 4.30i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.924 - 0.671i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.72 - 1.98i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.88 - 5.78i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.45T + 23T^{2} \)
29 \( 1 + (-1.02 - 3.15i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.44 - 1.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.460 - 1.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.539 - 1.66i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.263T + 43T^{2} \)
47 \( 1 + (2.13 - 6.58i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.16 - 0.846i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.18 - 6.72i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.02 - 1.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.516T + 67T^{2} \)
71 \( 1 + (8.68 - 6.30i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.75 + 5.40i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.14 + 6.64i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.62 + 2.63i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-2.71 - 1.97i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.45870879694471657692181493507, −9.800268679659781827183791751134, −8.862183445993965381632815645531, −7.64398775975671412484791812426, −6.98883385897856957673560850466, −6.20281896338120886299136715956, −4.66028917387182569044913587951, −4.15735686105579514874714016108, −3.39251280480858933203336832357, −1.42695882419103868011706777773, 0.60360615455217618450959993120, 2.19486593281202721575727165502, 3.26849738626738037655165044410, 4.64160958235582135237800050585, 5.73222203072802227658650288230, 6.48626244131401933772536551886, 7.11639968555678693470783487906, 8.431999788801859243851558678151, 8.938461718918580435236757543045, 9.617327176926765097256217090683

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