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Properties

Label 2-880-44.43-c1-0-17
Degree $2$
Conductor $880$
Sign $0.278 + 0.960i$
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  + 2.52i·3-s − 5-s − 2.20·7-s − 3.37·9-s + (−3.22 − 0.792i)11-s − 6.85i·13-s − 2.52i·15-s − 1.27i·17-s + 3.22·19-s − 5.57i·21-s − 1.58i·23-s + 25-s − 0.939i·27-s − 3.02i·29-s − 0.644i·31-s + ⋯
L(s)  = 1  + 1.45i·3-s − 0.447·5-s − 0.835·7-s − 1.12·9-s + (−0.971 − 0.238i)11-s − 1.90i·13-s − 0.651i·15-s − 0.309i·17-s + 0.738·19-s − 1.21i·21-s − 0.330i·23-s + 0.200·25-s − 0.180i·27-s − 0.562i·29-s − 0.115i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.278 + 0.960i$
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: \((2,\ 880,\ (\ :1/2),\ 0.278 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374649 - 0.281400i\)
\(L(\frac12)\) \(\approx\) \(0.374649 - 0.281400i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.22 + 0.792i)T \)
good3 \( 1 - 2.52iT - 3T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
13 \( 1 + 6.85iT - 13T^{2} \)
17 \( 1 + 1.27iT - 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 + 1.58iT - 23T^{2} \)
29 \( 1 + 3.02iT - 29T^{2} \)
31 \( 1 + 0.644iT - 31T^{2} \)
37 \( 1 + 0.372T + 37T^{2} \)
41 \( 1 - 5.10iT - 41T^{2} \)
43 \( 1 + 5.43T + 43T^{2} \)
47 \( 1 - 8.51iT - 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 8.51iT - 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 9.45iT - 71T^{2} \)
73 \( 1 + 1.75iT - 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
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

−9.861268694791590684617335049652, −9.527087455179744899684260972475, −8.240910655313764034947512445300, −7.72385204985637490258417246278, −6.29553165070098953017554522323, −5.32508535831674248484294335159, −4.68428091215558674452002094835, −3.30109661369858049647199902336, −3.06078713004679248616550316646, −0.22304086073103609776481134145, 1.51027918532451661256282618597, 2.62219351666854908836669212982, 3.84454047682586140051828209345, 5.16188536464514718410833188520, 6.33205072378582763991892559566, 6.97147416495114535175667123271, 7.53249323367381319645912956414, 8.488640463878387909551243249348, 9.348867756606788576193119031244, 10.30882327607498858499267724808

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