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Properties

Label 2-560-16.5-c1-0-10
Degree $2$
Conductor $560$
Sign $-0.0645 - 0.997i$
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  + (−0.914 − 1.07i)2-s + (−1.42 + 1.42i)3-s + (−0.326 + 1.97i)4-s + (0.707 + 0.707i)5-s + (2.84 + 0.234i)6-s + i·7-s + (2.42 − 1.45i)8-s − 1.07i·9-s + (0.115 − 1.40i)10-s + (1.52 + 1.52i)11-s + (−2.35 − 3.28i)12-s + (3.04 − 3.04i)13-s + (1.07 − 0.914i)14-s − 2.01·15-s + (−3.78 − 1.28i)16-s + 4.23·17-s + ⋯
L(s)  = 1  + (−0.646 − 0.762i)2-s + (−0.824 + 0.824i)3-s + (−0.163 + 0.986i)4-s + (0.316 + 0.316i)5-s + (1.16 + 0.0955i)6-s + 0.377i·7-s + (0.858 − 0.513i)8-s − 0.359i·9-s + (0.0366 − 0.445i)10-s + (0.459 + 0.459i)11-s + (−0.678 − 0.948i)12-s + (0.844 − 0.844i)13-s + (0.288 − 0.244i)14-s − 0.521·15-s + (−0.946 − 0.322i)16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0645 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0645 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.0645 - 0.997i$
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: \((2,\ 560,\ (\ :1/2),\ -0.0645 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481859 + 0.514048i\)
\(L(\frac12)\) \(\approx\) \(0.481859 + 0.514048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.914 + 1.07i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (1.42 - 1.42i)T - 3iT^{2} \)
11 \( 1 + (-1.52 - 1.52i)T + 11iT^{2} \)
13 \( 1 + (-3.04 + 3.04i)T - 13iT^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 + (4.95 - 4.95i)T - 19iT^{2} \)
23 \( 1 - 0.679iT - 23T^{2} \)
29 \( 1 + (6.70 - 6.70i)T - 29iT^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
37 \( 1 + (2.96 + 2.96i)T + 37iT^{2} \)
41 \( 1 - 7.19iT - 41T^{2} \)
43 \( 1 + (-5.87 - 5.87i)T + 43iT^{2} \)
47 \( 1 + 9.08T + 47T^{2} \)
53 \( 1 + (-0.694 - 0.694i)T + 53iT^{2} \)
59 \( 1 + (1.02 + 1.02i)T + 59iT^{2} \)
61 \( 1 + (-3.24 + 3.24i)T - 61iT^{2} \)
67 \( 1 + (8.40 - 8.40i)T - 67iT^{2} \)
71 \( 1 - 9.32iT - 71T^{2} \)
73 \( 1 + 7.80iT - 73T^{2} \)
79 \( 1 - 7.64T + 79T^{2} \)
83 \( 1 + (7.17 - 7.17i)T - 83iT^{2} \)
89 \( 1 + 12.8iT - 89T^{2} \)
97 \( 1 - 1.52T + 97T^{2} \)
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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

−10.86497892965792798959119225614, −10.18907673170369700080671927352, −9.631301562358219514311708759658, −8.547283806450649585400244165443, −7.66723028300616678475952176141, −6.29879920969656732196855895081, −5.40977293570424541676254846961, −4.18669084783049953519927713637, −3.20452915297946412572860484390, −1.60773564215633156188314638465, 0.59235174432903632918079397810, 1.75990711105118601626526734197, 4.09317421253520947829088173374, 5.38869816330746001896354043810, 6.25820763987036510912040747890, 6.72697371144373887893930864543, 7.72896042604594026187533937699, 8.745327180362925832779119657198, 9.434240961027842137895182111346, 10.57750546212924351289228407010

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