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Properties

Label 2-880-55.32-c1-0-19
Degree $2$
Conductor $880$
Sign $0.978 + 0.207i$
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  + (−1.70 + 1.70i)3-s + (−0.707 + 2.12i)5-s + (2.12 − 2.12i)7-s − 2.82i·9-s + (1.41 − 3i)11-s + (−3 − 3i)13-s + (−2.41 − 4.82i)15-s + (5.12 − 5.12i)17-s − 3·19-s + 7.24i·21-s + (0.171 − 0.171i)23-s + (−3.99 − 3i)25-s + (−0.292 − 0.292i)27-s + 1.24·29-s + 7.24·31-s + ⋯
L(s)  = 1  + (−0.985 + 0.985i)3-s + (−0.316 + 0.948i)5-s + (0.801 − 0.801i)7-s − 0.942i·9-s + (0.426 − 0.904i)11-s + (−0.832 − 0.832i)13-s + (−0.623 − 1.24i)15-s + (1.24 − 1.24i)17-s − 0.688·19-s + 1.58i·21-s + (0.0357 − 0.0357i)23-s + (−0.799 − 0.600i)25-s + (−0.0563 − 0.0563i)27-s + 0.230·29-s + 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979511 - 0.102578i\)
\(L(\frac12)\) \(\approx\) \(0.979511 - 0.102578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 2.12i)T \)
11 \( 1 + (-1.41 + 3i)T \)
good3 \( 1 + (1.70 - 1.70i)T - 3iT^{2} \)
7 \( 1 + (-2.12 + 2.12i)T - 7iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (-5.12 + 5.12i)T - 17iT^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-0.171 + 0.171i)T - 23iT^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 + (-0.121 - 0.121i)T + 37iT^{2} \)
41 \( 1 - 1.75iT - 41T^{2} \)
43 \( 1 + (-1.24 - 1.24i)T + 43iT^{2} \)
47 \( 1 + (4.41 + 4.41i)T + 47iT^{2} \)
53 \( 1 + (-9.53 + 9.53i)T - 53iT^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 - 7.24iT - 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (7.24 + 7.24i)T + 83iT^{2} \)
89 \( 1 - 5.48iT - 89T^{2} \)
97 \( 1 + (2.24 + 2.24i)T + 97iT^{2} \)
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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

−10.28579926623106580827191754703, −9.739427736940178006268992287345, −8.281741812066952630818560613144, −7.54194834985726014603808613973, −6.60204485530820920592888676283, −5.58055815458882746690506514246, −4.81062179527218342776315664922, −3.88836829419800040627736164487, −2.85678901668547518258239522610, −0.62036459299959450696983813829, 1.24991673074454479787185590456, 2.08347937076167921290407390038, 4.17178525684872472893920718355, 4.99417860622411293642445655385, 5.79639053376877714650266302284, 6.67724260205588079154329630437, 7.64086308166406628007243959183, 8.314864136182213212673839239709, 9.237136504161169172891030396134, 10.20600342620145185713160874130

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