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Properties

Label 2-880-11.5-c1-0-3
Degree $2$
Conductor $880$
Sign $-0.800 - 0.599i$
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  + (1.54 + 1.12i)3-s + (−0.309 + 0.951i)5-s + (−2.48 + 1.80i)7-s + (0.203 + 0.627i)9-s + (−1.86 + 2.74i)11-s + (0.942 + 2.90i)13-s + (−1.54 + 1.12i)15-s + (−0.143 + 0.441i)17-s + (−6.38 − 4.64i)19-s − 5.86·21-s + 1.39·23-s + (−0.809 − 0.587i)25-s + (1.38 − 4.25i)27-s + (−3.01 + 2.18i)29-s + (3.23 + 9.96i)31-s + ⋯
L(s)  = 1  + (0.893 + 0.649i)3-s + (−0.138 + 0.425i)5-s + (−0.937 + 0.681i)7-s + (0.0679 + 0.209i)9-s + (−0.561 + 0.827i)11-s + (0.261 + 0.804i)13-s + (−0.399 + 0.290i)15-s + (−0.0347 + 0.106i)17-s + (−1.46 − 1.06i)19-s − 1.28·21-s + 0.289·23-s + (−0.161 − 0.117i)25-s + (0.266 − 0.819i)27-s + (−0.559 + 0.406i)29-s + (0.581 + 1.78i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413473 + 1.24279i\)
\(L(\frac12)\) \(\approx\) \(0.413473 + 1.24279i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (1.86 - 2.74i)T \)
good3 \( 1 + (-1.54 - 1.12i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.48 - 1.80i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.942 - 2.90i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.143 - 0.441i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.38 + 4.64i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 + (3.01 - 2.18i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.23 - 9.96i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.49 - 1.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.56 - 2.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 + (2.41 + 1.75i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.29 - 3.98i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.27 + 1.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.623 - 1.91i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 6.75T + 67T^{2} \)
71 \( 1 + (-2.01 + 6.20i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.98 + 5.80i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.57 - 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.75 - 8.48i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (-4.74 - 14.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.33176693134907821719198287530, −9.454196425095217227849567009295, −8.982781394725176826961122051265, −8.204460485152300122463997683866, −6.90471177739067980854266155561, −6.40883153653989490037192324767, −4.98698658711277762614958408930, −4.03131581987928409548210357523, −3.03462462009339411885213971764, −2.27273727117260405468084850338, 0.52944999379856866112900339289, 2.18177667691331622909892891515, 3.25853473679431278766848164245, 4.10300153941495192293538363988, 5.58634516567102065907399143904, 6.39560491753251560060213091087, 7.51766603243495192880304419136, 8.104977994296845204408332215180, 8.720583087941580169052985585144, 9.779975152684366051919723947254

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