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Properties

Label 2-880-55.43-c1-0-21
Degree $2$
Conductor $880$
Sign $0.248 + 0.968i$
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  + (−0.395 − 0.395i)3-s + (−0.0577 − 2.23i)5-s + (2.57 + 2.57i)7-s − 2.68i·9-s + (2.35 − 2.33i)11-s + (−1.81 + 1.81i)13-s + (−0.861 + 0.906i)15-s + (−2.63 − 2.63i)17-s + 4.84·19-s − 2.03i·21-s + (0.648 + 0.648i)23-s + (−4.99 + 0.258i)25-s + (−2.24 + 2.24i)27-s + 9.19·29-s − 9.88·31-s + ⋯
L(s)  = 1  + (−0.228 − 0.228i)3-s + (−0.0258 − 0.999i)5-s + (0.973 + 0.973i)7-s − 0.895i·9-s + (0.711 − 0.702i)11-s + (−0.502 + 0.502i)13-s + (−0.222 + 0.234i)15-s + (−0.640 − 0.640i)17-s + 1.11·19-s − 0.444i·21-s + (0.135 + 0.135i)23-s + (−0.998 + 0.0516i)25-s + (−0.432 + 0.432i)27-s + 1.70·29-s − 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.248 + 0.968i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22769 - 0.952258i\)
\(L(\frac12)\) \(\approx\) \(1.22769 - 0.952258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0577 + 2.23i)T \)
11 \( 1 + (-2.35 + 2.33i)T \)
good3 \( 1 + (0.395 + 0.395i)T + 3iT^{2} \)
7 \( 1 + (-2.57 - 2.57i)T + 7iT^{2} \)
13 \( 1 + (1.81 - 1.81i)T - 13iT^{2} \)
17 \( 1 + (2.63 + 2.63i)T + 17iT^{2} \)
19 \( 1 - 4.84T + 19T^{2} \)
23 \( 1 + (-0.648 - 0.648i)T + 23iT^{2} \)
29 \( 1 - 9.19T + 29T^{2} \)
31 \( 1 + 9.88T + 31T^{2} \)
37 \( 1 + (-7.37 + 7.37i)T - 37iT^{2} \)
41 \( 1 + 5.37iT - 41T^{2} \)
43 \( 1 + (2.39 - 2.39i)T - 43iT^{2} \)
47 \( 1 + (-2.29 + 2.29i)T - 47iT^{2} \)
53 \( 1 + (3.34 + 3.34i)T + 53iT^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + 6.46iT - 61T^{2} \)
67 \( 1 + (-0.909 + 0.909i)T - 67iT^{2} \)
71 \( 1 - 5.95T + 71T^{2} \)
73 \( 1 + (-1.99 + 1.99i)T - 73iT^{2} \)
79 \( 1 + 7.61T + 79T^{2} \)
83 \( 1 + (4.50 - 4.50i)T - 83iT^{2} \)
89 \( 1 - 2.85iT - 89T^{2} \)
97 \( 1 + (8.73 - 8.73i)T - 97iT^{2} \)
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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

−9.541078898353642754315885322202, −9.140409237284134156829193039892, −8.477217306082113543602167085388, −7.46228243445745384955676225054, −6.40842032936986553205200272885, −5.47302099829614784241228230840, −4.81020641706679037661964008731, −3.63272027199325003653425434052, −2.09257852035320741803609276292, −0.845438490315405683130615648183, 1.54323392775889711349739453845, 2.84633538232839064674259610019, 4.19828955401465451135937291576, 4.81061146188552581242997666233, 6.00316984303716920937004103725, 7.16674537141498159793285291438, 7.52754012875105515827192339572, 8.483347535436604543761701943051, 9.883856577934323842706726898405, 10.28959461007637436096882280951

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