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Properties

Label 2-560-140.139-c1-0-9
Degree $2$
Conductor $560$
Sign $-i$
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  + 2.64i·3-s + 2.23·5-s − 2.64i·7-s − 4.00·9-s + 5.91i·11-s + 6.70·13-s + 5.91i·15-s − 2.23·17-s + 7.00·21-s + 5.00·25-s − 2.64i·27-s − 9·29-s − 15.6·33-s − 5.91i·35-s + 17.7i·39-s + ⋯
L(s)  = 1  + 1.52i·3-s + 0.999·5-s − 0.999i·7-s − 1.33·9-s + 1.78i·11-s + 1.86·13-s + 1.52i·15-s − 0.542·17-s + 1.52·21-s + 1.00·25-s − 0.509i·27-s − 1.67·29-s − 2.72·33-s − 0.999i·35-s + 2.84i·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23584 + 1.23584i\)
\(L(\frac12)\) \(\approx\) \(1.23584 + 1.23584i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
7 \( 1 + 2.64iT \)
good3 \( 1 - 2.64iT - 3T^{2} \)
11 \( 1 - 5.91iT - 11T^{2} \)
13 \( 1 - 6.70T + 13T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 7.93iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + 17.7iT - 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 6.70T + 97T^{2} \)
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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

−10.66954090844931031590872383475, −10.13913881818492534941844327537, −9.445838282390065584019552088799, −8.737634963657083135362414403344, −7.32759331364956190110273814810, −6.30732727850357389590641394914, −5.20743432341937137789442159068, −4.31718959671131993211945534361, −3.56381427615698403380593739179, −1.80781462286325931328285300950, 1.14028031522247428873943626076, 2.22772490559129737407803012393, 3.39285386945340527592384234203, 5.56243038409500412092873899140, 6.04201329549139387831827936644, 6.61489566957039568281396281487, 8.041852891433356914165428215387, 8.655841145447821324806270587564, 9.319701031157154935781007869406, 11.02590022911108232670072797790

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