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Properties

Label 2-880-11.5-c1-0-17
Degree $2$
Conductor $880$
Sign $0.652 + 0.758i$
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  + (0.582 + 0.422i)3-s + (−0.309 + 0.951i)5-s + (3.38 − 2.45i)7-s + (−0.767 − 2.36i)9-s + (−2.41 + 2.27i)11-s + (−1.86 − 5.75i)13-s + (−0.582 + 0.422i)15-s + (2.01 − 6.20i)17-s + (0.598 + 0.434i)19-s + 3.01·21-s + 5.13·23-s + (−0.809 − 0.587i)25-s + (1.21 − 3.75i)27-s + (−1.68 + 1.22i)29-s + (0.348 + 1.07i)31-s + ⋯
L(s)  = 1  + (0.336 + 0.244i)3-s + (−0.138 + 0.425i)5-s + (1.27 − 0.929i)7-s + (−0.255 − 0.786i)9-s + (−0.727 + 0.686i)11-s + (−0.518 − 1.59i)13-s + (−0.150 + 0.109i)15-s + (0.488 − 1.50i)17-s + (0.137 + 0.0997i)19-s + 0.657·21-s + 1.06·23-s + (−0.161 − 0.117i)25-s + (0.234 − 0.722i)27-s + (−0.312 + 0.227i)29-s + (0.0625 + 0.192i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.652 + 0.758i$
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: \((2,\ 880,\ (\ :1/2),\ 0.652 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58626 - 0.727841i\)
\(L(\frac12)\) \(\approx\) \(1.58626 - 0.727841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 - 2.27i)T \)
good3 \( 1 + (-0.582 - 0.422i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-3.38 + 2.45i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.86 + 5.75i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.01 + 6.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.598 - 0.434i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.13T + 23T^{2} \)
29 \( 1 + (1.68 - 1.22i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.348 - 1.07i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.75 - 4.18i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.25 - 2.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.29T + 43T^{2} \)
47 \( 1 + (-7.47 - 5.43i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.40 - 7.38i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.98 + 3.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.53 + 13.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.53T + 67T^{2} \)
71 \( 1 + (2.74 - 8.43i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.54 + 4.75i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.176 - 0.542i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.70 + 14.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.33T + 89T^{2} \)
97 \( 1 + (-3.07 - 9.47i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.13792837364093733746771430785, −9.286562369387877004755627768712, −8.100387500968807967149827407787, −7.59693123171105100978775462223, −6.87986592205511703249113999262, −5.31238181511774780184331274604, −4.81125173617394953096756642075, −3.48804512818210316982992866616, −2.64505738687594244955938383382, −0.847435964789743550745600213937, 1.68403645475564312576439353745, 2.45136883825335386623294010873, 4.00441703234025031417956659653, 5.14543855580491335843071385946, 5.57506410491336356031720755297, 7.01944629373919885475099292008, 7.957319430565084941940268481663, 8.550860398178052068059370273543, 9.035676302575786210648255160556, 10.37691802098720086999250262946

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