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Properties

Label 2-560-16.5-c1-0-21
Degree $2$
Conductor $560$
Sign $0.891 + 0.453i$
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  + (−1.16 − 0.808i)2-s + (0.925 − 0.925i)3-s + (0.692 + 1.87i)4-s + (−0.707 − 0.707i)5-s + (−1.82 + 0.325i)6-s + i·7-s + (0.713 − 2.73i)8-s + 1.28i·9-s + (0.248 + 1.39i)10-s + (3.01 + 3.01i)11-s + (2.37 + 1.09i)12-s + (−0.350 + 0.350i)13-s + (0.808 − 1.16i)14-s − 1.30·15-s + (−3.04 + 2.59i)16-s + 5.90·17-s + ⋯
L(s)  = 1  + (−0.820 − 0.571i)2-s + (0.534 − 0.534i)3-s + (0.346 + 0.938i)4-s + (−0.316 − 0.316i)5-s + (−0.744 + 0.132i)6-s + 0.377i·7-s + (0.252 − 0.967i)8-s + 0.428i·9-s + (0.0786 + 0.440i)10-s + (0.908 + 0.908i)11-s + (0.686 + 0.316i)12-s + (−0.0971 + 0.0971i)13-s + (0.216 − 0.310i)14-s − 0.338·15-s + (−0.760 + 0.649i)16-s + 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16168 - 0.278689i\)
\(L(\frac12)\) \(\approx\) \(1.16168 - 0.278689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.808i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.925 + 0.925i)T - 3iT^{2} \)
11 \( 1 + (-3.01 - 3.01i)T + 11iT^{2} \)
13 \( 1 + (0.350 - 0.350i)T - 13iT^{2} \)
17 \( 1 - 5.90T + 17T^{2} \)
19 \( 1 + (-0.223 + 0.223i)T - 19iT^{2} \)
23 \( 1 + 2.20iT - 23T^{2} \)
29 \( 1 + (-1.25 + 1.25i)T - 29iT^{2} \)
31 \( 1 - 2.03T + 31T^{2} \)
37 \( 1 + (1.71 + 1.71i)T + 37iT^{2} \)
41 \( 1 - 8.96iT - 41T^{2} \)
43 \( 1 + (5.64 + 5.64i)T + 43iT^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + (1.26 + 1.26i)T + 53iT^{2} \)
59 \( 1 + (3.11 + 3.11i)T + 59iT^{2} \)
61 \( 1 + (-2.39 + 2.39i)T - 61iT^{2} \)
67 \( 1 + (-4.05 + 4.05i)T - 67iT^{2} \)
71 \( 1 + 9.47iT - 71T^{2} \)
73 \( 1 - 6.05iT - 73T^{2} \)
79 \( 1 + 1.37T + 79T^{2} \)
83 \( 1 + (-8.19 + 8.19i)T - 83iT^{2} \)
89 \( 1 - 9.38iT - 89T^{2} \)
97 \( 1 + 6.87T + 97T^{2} \)
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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

−10.56175606807295625916208155130, −9.717536529840311368837501294457, −8.936097595257397049919860280653, −8.088270131005028125822928668348, −7.48246464302477387914620937388, −6.52872756242829804902478879357, −4.90260363910088143758833072080, −3.65001157975866531231544830647, −2.42723272656663324228562430980, −1.32684929482575652865582319902, 1.04572828972344922312156025646, 3.06096156124269589220939911827, 4.01784304377021623141057906596, 5.50389962599408663993287821834, 6.46472818375848320574626535277, 7.37816057543397858857072609687, 8.297503222151617522490409541484, 9.017575677437447334013768628819, 9.838607826838600194272504386929, 10.49405217908605683703261794567

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