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Properties

Label 2-880-11.3-c1-0-1
Degree $2$
Conductor $880$
Sign $-0.980 + 0.197i$
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  + (0.507 + 1.56i)3-s + (−0.809 − 0.587i)5-s + (−1.04 + 3.21i)7-s + (0.248 − 0.180i)9-s + (−3.04 − 1.30i)11-s + (−3.81 + 2.76i)13-s + (0.507 − 1.56i)15-s + (−6.11 − 4.44i)17-s + (0.689 + 2.12i)19-s − 5.54·21-s + 0.0373·23-s + (0.309 + 0.951i)25-s + (4.39 + 3.18i)27-s + (1.96 − 6.05i)29-s + (−0.282 + 0.205i)31-s + ⋯
L(s)  = 1  + (0.292 + 0.901i)3-s + (−0.361 − 0.262i)5-s + (−0.394 + 1.21i)7-s + (0.0827 − 0.0600i)9-s + (−0.918 − 0.394i)11-s + (−1.05 + 0.767i)13-s + (0.130 − 0.402i)15-s + (−1.48 − 1.07i)17-s + (0.158 + 0.486i)19-s − 1.20·21-s + 0.00778·23-s + (0.0618 + 0.190i)25-s + (0.844 + 0.613i)27-s + (0.365 − 1.12i)29-s + (−0.0506 + 0.0368i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.980 + 0.197i$
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: \((2,\ 880,\ (\ :1/2),\ -0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0500176 - 0.500780i\)
\(L(\frac12)\) \(\approx\) \(0.0500176 - 0.500780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 + 1.30i)T \)
good3 \( 1 + (-0.507 - 1.56i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.04 - 3.21i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.81 - 2.76i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (6.11 + 4.44i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.689 - 2.12i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.0373T + 23T^{2} \)
29 \( 1 + (-1.96 + 6.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.282 - 0.205i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.04 - 6.30i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.609 + 1.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + (-1.96 - 6.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.33 - 5.32i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.959 + 2.95i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-9.45 - 6.86i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.94T + 67T^{2} \)
71 \( 1 + (-0.193 - 0.140i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.12 + 12.7i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.73 - 4.16i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.72 + 7.06i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
97 \( 1 + (-12.7 + 9.27i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.35808546970270791077034253165, −9.622765839047230419818702767006, −9.047924278608207325385538426003, −8.333646208937685193809146542588, −7.20877329573012308877375407522, −6.21422634834603047295036704510, −5.00395166876185452170622471892, −4.52769358132351094052898026091, −3.18660098460678539548148563415, −2.32627584686628305733340183599, 0.21457458987669847359422425849, 1.91934842880572969199580752195, 3.01815823358157553697475345979, 4.23502636896884036877514368323, 5.18336306944197557306224588961, 6.81245820452556903741666766858, 6.99458065409367561713590181629, 7.85879044261877062601434097813, 8.552416369929619678863592151556, 9.976155962383060405331285683737

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