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Properties

Label 2-880-220.43-c0-0-5
Degree $2$
Conductor $880$
Sign $-0.0299 + 0.999i$
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  + (1.36 − 1.36i)3-s + (−0.866 − 0.5i)5-s − 2.73i·9-s + i·11-s + (−1.86 + 0.499i)15-s + (−0.366 + 0.366i)23-s + (0.499 + 0.866i)25-s + (−2.36 − 2.36i)27-s + i·31-s + (1.36 + 1.36i)33-s + (1.36 − 1.36i)37-s + (−1.36 + 2.36i)45-s + (1 + i)47-s + i·49-s + (−1 − i)53-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)3-s + (−0.866 − 0.5i)5-s − 2.73i·9-s + i·11-s + (−1.86 + 0.499i)15-s + (−0.366 + 0.366i)23-s + (0.499 + 0.866i)25-s + (−2.36 − 2.36i)27-s + i·31-s + (1.36 + 1.36i)33-s + (1.36 − 1.36i)37-s + (−1.36 + 2.36i)45-s + (1 + i)47-s + i·49-s + (−1 − i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.0299 + 0.999i$
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: \((2,\ 880,\ (\ :0),\ -0.0299 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.307318746\)
\(L(\frac12)\) \(\approx\) \(1.307318746\)
\(L(1)\) 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 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
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 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
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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

−9.738784294800882116945047082011, −9.065960405605261693077830906208, −8.318036531506375984359046614845, −7.54060717930936533961767526011, −7.19248134360475378628894512528, −6.05658600125537919109297168332, −4.51015432566838211828285673862, −3.56159649848261142436627024358, −2.47176325094174984925014367806, −1.30898036881670797950025826654, 2.50056518450525596810840107745, 3.32592014572024620001501582188, 4.03988941904665264019732286370, 4.88540701164703095623888673719, 6.22094143731718373599574831723, 7.63527527492659467278258616601, 8.128681149053273622312744640116, 8.860926954601703184616761703387, 9.668442940790191879555065636418, 10.50776876473902676792622230991

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