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Properties

Label 2-880-44.43-c1-0-14
Degree $2$
Conductor $880$
Sign $0.334 + 0.942i$
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  + 0.792i·3-s − 5-s − 4.70·7-s + 2.37·9-s + (2.15 − 2.52i)11-s − 1.01i·13-s − 0.792i·15-s + 2.71i·17-s − 2.15·19-s − 3.72i·21-s − 5.04i·23-s + 25-s + 4.25i·27-s − 9.15i·29-s − 9.30i·31-s + ⋯
L(s)  = 1  + 0.457i·3-s − 0.447·5-s − 1.77·7-s + 0.790·9-s + (0.648 − 0.761i)11-s − 0.280i·13-s − 0.204i·15-s + 0.658i·17-s − 0.493·19-s − 0.813i·21-s − 1.05i·23-s + 0.200·25-s + 0.819i·27-s − 1.70i·29-s − 1.67i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.334 + 0.942i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.334 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755335 - 0.533198i\)
\(L(\frac12)\) \(\approx\) \(0.755335 - 0.533198i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + (-2.15 + 2.52i)T \)
good3 \( 1 - 0.792iT - 3T^{2} \)
7 \( 1 + 4.70T + 7T^{2} \)
13 \( 1 + 1.01iT - 13T^{2} \)
17 \( 1 - 2.71iT - 17T^{2} \)
19 \( 1 + 2.15T + 19T^{2} \)
23 \( 1 + 5.04iT - 23T^{2} \)
29 \( 1 + 9.15iT - 29T^{2} \)
31 \( 1 + 9.30iT - 31T^{2} \)
37 \( 1 - 5.37T + 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 + 2.55T + 43T^{2} \)
47 \( 1 + 1.87iT - 47T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 - 1.87iT - 59T^{2} \)
61 \( 1 - 7.13iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 6.13iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + 8.60T + 79T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.996023561586294321259487353964, −9.268912934619479470514191900042, −8.435380844179165304308039581306, −7.33230449924978203294201152913, −6.41130544187989628447536198774, −5.85077068205718498172721866198, −4.14353301734671952980737443013, −3.84148818266911029627868883947, −2.58540431174091890930726674019, −0.47569391047832212090069128189, 1.37816148576462315052239913049, 2.96120745382842363113386827532, 3.85106590707308552380405140457, 4.91673689360451725952396718622, 6.36168581037502560989863687455, 6.87680909630635646685739180702, 7.44389508529117203071597252616, 8.759248086594311030056719766684, 9.588569425651617546852055661050, 10.03471231515692327259933360752

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