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Properties

Label 2-560-28.19-c1-0-8
Degree $2$
Conductor $560$
Sign $0.963 - 0.269i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (0.821 + 1.42i)3-s + (−0.866 − 0.5i)5-s + (1.59 − 2.10i)7-s + (0.149 − 0.259i)9-s + (3.12 − 1.80i)11-s + 6.16i·13-s − 1.64i·15-s + (3.07 − 1.77i)17-s + (1.14 − 1.99i)19-s + (4.31 + 0.539i)21-s + (−4.75 − 2.74i)23-s + (0.499 + 0.866i)25-s + 5.42·27-s + 3.52·29-s + (2.33 + 4.04i)31-s + ⋯
L(s)  = 1  + (0.474 + 0.821i)3-s + (−0.387 − 0.223i)5-s + (0.603 − 0.797i)7-s + (0.0499 − 0.0864i)9-s + (0.941 − 0.543i)11-s + 1.70i·13-s − 0.424i·15-s + (0.745 − 0.430i)17-s + (0.263 − 0.456i)19-s + (0.941 + 0.117i)21-s + (−0.991 − 0.572i)23-s + (0.0999 + 0.173i)25-s + 1.04·27-s + 0.655·29-s + (0.419 + 0.725i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.963 - 0.269i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.963 - 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80394 + 0.247342i\)
\(L(\frac12)\) \(\approx\) \(1.80394 + 0.247342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-1.59 + 2.10i)T \)
good3 \( 1 + (-0.821 - 1.42i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.16iT - 13T^{2} \)
17 \( 1 + (-3.07 + 1.77i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 + 1.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.75 + 2.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
31 \( 1 + (-2.33 - 4.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.44 - 7.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.04iT - 41T^{2} \)
43 \( 1 + 6.61iT - 43T^{2} \)
47 \( 1 + (-0.614 + 1.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.22 - 7.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.91 - 3.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.6 + 6.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.95 - 4.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + (6.30 - 3.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.73 + 2.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + (13.3 + 7.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.17iT - 97T^{2} \)
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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

−10.68147787407504202433787025923, −9.874752813478925807622249061301, −9.017318494936328110437028119587, −8.410377465995568867032934009468, −7.20900737112386483345234983362, −6.39659669307857029321057471332, −4.71464619909598866585574932967, −4.22386782920550930178385219821, −3.25985634099305432985304590951, −1.32531883068709721480998881047, 1.43525304341408430340884694717, 2.62694972534347894566073290309, 3.86874588863043863361316122501, 5.27551908702780037081832961805, 6.18360759977487294790773561449, 7.49278109123466372926366526298, 7.898671129373055290751404119124, 8.684158520084294122736870364900, 9.882026574315967705417397547482, 10.69989059150996576204874269637

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