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Properties

Label 2-880-55.43-c1-0-7
Degree $2$
Conductor $880$
Sign $0.876 - 0.481i$
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  + (0.185 + 0.185i)3-s + (−2.05 + 0.892i)5-s + (0.202 + 0.202i)7-s − 2.93i·9-s + (2.64 − 1.99i)11-s + (−3.54 + 3.54i)13-s + (−0.546 − 0.214i)15-s + (2.99 + 2.99i)17-s + 7.13·19-s + 0.0752i·21-s + (3.44 + 3.44i)23-s + (3.40 − 3.65i)25-s + (1.10 − 1.10i)27-s − 3.59·29-s + 6.71·31-s + ⋯
L(s)  = 1  + (0.107 + 0.107i)3-s + (−0.916 + 0.399i)5-s + (0.0765 + 0.0765i)7-s − 0.977i·9-s + (0.798 − 0.601i)11-s + (−0.984 + 0.984i)13-s + (−0.141 − 0.0554i)15-s + (0.725 + 0.725i)17-s + 1.63·19-s + 0.0164i·21-s + (0.718 + 0.718i)23-s + (0.681 − 0.731i)25-s + (0.211 − 0.211i)27-s − 0.667·29-s + 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.876 - 0.481i$
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: \((2,\ 880,\ (\ :1/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38671 + 0.355758i\)
\(L(\frac12)\) \(\approx\) \(1.38671 + 0.355758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 - 0.892i)T \)
11 \( 1 + (-2.64 + 1.99i)T \)
good3 \( 1 + (-0.185 - 0.185i)T + 3iT^{2} \)
7 \( 1 + (-0.202 - 0.202i)T + 7iT^{2} \)
13 \( 1 + (3.54 - 3.54i)T - 13iT^{2} \)
17 \( 1 + (-2.99 - 2.99i)T + 17iT^{2} \)
19 \( 1 - 7.13T + 19T^{2} \)
23 \( 1 + (-3.44 - 3.44i)T + 23iT^{2} \)
29 \( 1 + 3.59T + 29T^{2} \)
31 \( 1 - 6.71T + 31T^{2} \)
37 \( 1 + (-0.924 + 0.924i)T - 37iT^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (8.17 - 8.17i)T - 43iT^{2} \)
47 \( 1 + (-6.13 + 6.13i)T - 47iT^{2} \)
53 \( 1 + (-8.77 - 8.77i)T + 53iT^{2} \)
59 \( 1 - 4.44iT - 59T^{2} \)
61 \( 1 - 0.601iT - 61T^{2} \)
67 \( 1 + (-1.46 + 1.46i)T - 67iT^{2} \)
71 \( 1 + 8.47T + 71T^{2} \)
73 \( 1 + (-6.69 + 6.69i)T - 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (-3.13 + 3.13i)T - 83iT^{2} \)
89 \( 1 + 4.12iT - 89T^{2} \)
97 \( 1 + (-5.71 + 5.71i)T - 97iT^{2} \)
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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

−10.01905498927539528323236869956, −9.422026423450605010978999443696, −8.564620522864668689019961166015, −7.57710743591293837792825568651, −6.88873088225404164203795151194, −5.98435104275583384765608829610, −4.72099999556630695557289566745, −3.70057397520530897633864384304, −3.04146340362219182202494025987, −1.13094802073193870523743377204, 0.894804447717804948862847978006, 2.57040616433081619920369956250, 3.67697193410298168446559907221, 4.91478680048679533923964308859, 5.31252108743274498886415537822, 7.05982382335840725970918522311, 7.48734023739384712708779320218, 8.225494188836188613317599773598, 9.242709363475544465339084649939, 10.04481869760300572603101164612

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