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Properties

Label 2-880-80.3-c1-0-4
Degree $2$
Conductor $880$
Sign $-0.987 - 0.155i$
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  + (−1.28 + 0.595i)2-s − 0.00646·3-s + (1.29 − 1.52i)4-s + (1.81 − 1.30i)5-s + (0.00828 − 0.00384i)6-s + (−1.60 + 1.60i)7-s + (−0.744 + 2.72i)8-s − 2.99·9-s + (−1.55 + 2.75i)10-s + (0.707 + 0.707i)11-s + (−0.00833 + 0.00987i)12-s + 3.94i·13-s + (1.10 − 3.00i)14-s + (−0.0117 + 0.00840i)15-s + (−0.670 − 3.94i)16-s + (−1.82 + 1.82i)17-s + ⋯
L(s)  = 1  + (−0.906 + 0.421i)2-s − 0.00373·3-s + (0.645 − 0.764i)4-s + (0.813 − 0.582i)5-s + (0.00338 − 0.00157i)6-s + (−0.605 + 0.605i)7-s + (−0.263 + 0.964i)8-s − 0.999·9-s + (−0.492 + 0.870i)10-s + (0.213 + 0.213i)11-s + (−0.00240 + 0.00285i)12-s + 1.09i·13-s + (0.294 − 0.804i)14-s + (−0.00303 + 0.00217i)15-s + (−0.167 − 0.985i)16-s + (−0.441 + 0.441i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.987 - 0.155i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0197174 + 0.251340i\)
\(L(\frac12)\) \(\approx\) \(0.0197174 + 0.251340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.595i)T \)
5 \( 1 + (-1.81 + 1.30i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + 0.00646T + 3T^{2} \)
7 \( 1 + (1.60 - 1.60i)T - 7iT^{2} \)
13 \( 1 - 3.94iT - 13T^{2} \)
17 \( 1 + (1.82 - 1.82i)T - 17iT^{2} \)
19 \( 1 + (5.41 + 5.41i)T + 19iT^{2} \)
23 \( 1 + (2.47 + 2.47i)T + 23iT^{2} \)
29 \( 1 + (5.56 - 5.56i)T - 29iT^{2} \)
31 \( 1 - 0.407iT - 31T^{2} \)
37 \( 1 - 6.88iT - 37T^{2} \)
41 \( 1 - 2.33iT - 41T^{2} \)
43 \( 1 + 0.446iT - 43T^{2} \)
47 \( 1 + (8.15 + 8.15i)T + 47iT^{2} \)
53 \( 1 - 9.72T + 53T^{2} \)
59 \( 1 + (3.21 - 3.21i)T - 59iT^{2} \)
61 \( 1 + (3.02 + 3.02i)T + 61iT^{2} \)
67 \( 1 - 9.45iT - 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (5.87 - 5.87i)T - 73iT^{2} \)
79 \( 1 + 1.69T + 79T^{2} \)
83 \( 1 + 3.04T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 + (-1.72 + 1.72i)T - 97iT^{2} \)
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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

−10.29307132348280426967477262853, −9.442216228615016357737456981834, −8.730902042268550333154136848287, −8.568645119715328501712111191127, −6.92714007449293701178219478401, −6.35829832131934719259878771710, −5.61294618114873284941179454338, −4.55911573339113290609927546637, −2.68402042406239017964178921587, −1.79477329454608866152952657617, 0.14696662737646102259473012965, 1.96586279272811955025828117199, 3.00122676637887782341989695734, 3.86197322620054014931098236545, 5.78972159900916670539125289964, 6.25241744372567812471207225281, 7.35549912863375846671051302390, 8.126636418340056680385093246402, 9.085742800742379783027940920385, 9.818705657775696139699386613207

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