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Properties

Label 2-880-220.43-c0-0-1
Degree $2$
Conductor $880$
Sign $0.880 - 0.473i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 0.366i)3-s + (0.866 − 0.5i)5-s + 0.732i·9-s + i·11-s + (−0.133 + 0.5i)15-s + (1.36 − 1.36i)23-s + (0.499 − 0.866i)25-s + (−0.633 − 0.633i)27-s + i·31-s + (−0.366 − 0.366i)33-s + (−0.366 + 0.366i)37-s + (0.366 + 0.633i)45-s + (1 + i)47-s + i·49-s + (−1 − i)53-s + ⋯
L(s)  = 1  + (−0.366 + 0.366i)3-s + (0.866 − 0.5i)5-s + 0.732i·9-s + i·11-s + (−0.133 + 0.5i)15-s + (1.36 − 1.36i)23-s + (0.499 − 0.866i)25-s + (−0.633 − 0.633i)27-s + i·31-s + (−0.366 − 0.366i)33-s + (−0.366 + 0.366i)37-s + (0.366 + 0.633i)45-s + (1 + i)47-s + i·49-s + (−1 − i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :0),\ 0.880 - 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024120888\)
\(L(\frac12)\) \(\approx\) \(1.024120888\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 - iT \)
good3 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
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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

−10.53300975140683356223233230199, −9.580456925980598883118975641790, −8.935047605688788564916457467928, −7.921191323776663331457390067248, −6.88830834718349923319945872170, −5.99249959261409142109989320512, −4.86204698256454426250430980842, −4.63768991866909904581888063658, −2.83074672534406765355507858593, −1.65509299013494404753012030906, 1.29913537355058552031294107842, 2.79721978330171152997967239463, 3.75478048464613548416210600467, 5.37159322268994136238573860278, 5.91139212460404454348784898472, 6.77605229499364546240785989781, 7.50795037723133448432457129126, 8.829010455394283372519730332327, 9.374902958869219581146743313046, 10.32047398856824109559921579938

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