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Properties

Label 2-560-140.67-c1-0-7
Degree $2$
Conductor $560$
Sign $0.164 - 0.986i$
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  + (1.67 − 0.448i)3-s + (−1.23 + 1.86i)5-s + (−1.48 + 2.19i)7-s + (2.12 + 1.22i)11-s + (−2 + 2i)13-s + (−1.22 + 3.67i)15-s + (2.73 − 0.732i)17-s + (1.22 + 2.12i)19-s + (−1.50 + 4.33i)21-s + (−1.34 + 5.01i)23-s + (−1.96 − 4.59i)25-s + (−3.67 + 3.67i)27-s − i·29-s + (4.24 + 2.44i)31-s + (4.09 + 1.09i)33-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)3-s + (−0.550 + 0.834i)5-s + (−0.560 + 0.827i)7-s + (0.639 + 0.369i)11-s + (−0.554 + 0.554i)13-s + (−0.316 + 0.948i)15-s + (0.662 − 0.177i)17-s + (0.280 + 0.486i)19-s + (−0.327 + 0.944i)21-s + (−0.280 + 1.04i)23-s + (−0.392 − 0.919i)25-s + (−0.707 + 0.707i)27-s − 0.185i·29-s + (0.762 + 0.439i)31-s + (0.713 + 0.191i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17339 + 0.994137i\)
\(L(\frac12)\) \(\approx\) \(1.17339 + 0.994137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 1.86i)T \)
7 \( 1 + (1.48 - 2.19i)T \)
good3 \( 1 + (-1.67 + 0.448i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.12 - 1.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (-2.73 + 0.732i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.22 - 2.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.34 - 5.01i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + iT - 29T^{2} \)
31 \( 1 + (-4.24 - 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 + 10.9i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 + (-6.12 - 6.12i)T + 43iT^{2} \)
47 \( 1 + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.12 + 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.448 + 1.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + (-0.732 - 2.73i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.57 - 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.67 + 3.67i)T + 83iT^{2} \)
89 \( 1 + (9.52 - 5.5i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11 - 11i)T + 97iT^{2} \)
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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

−11.06485796853677008685119110709, −9.730359265174143255088929296905, −9.337480713062868540832502314311, −8.187725753779859383674908214749, −7.50821272903748064026256680237, −6.62098410614595402617247483182, −5.53495101802791738568183865204, −3.94610996556922463831982860764, −3.05853000600049952973930822635, −2.11048668010361665783309347325, 0.800575135563416237115623434402, 2.84633774989759856270426917601, 3.76683698701370156428268347533, 4.61924539820769730787595521372, 5.98386668993535461735437603186, 7.19852173375136012067751979964, 8.096114746597406150375811265929, 8.731227927339443977020313520165, 9.620883261619730078951046765370, 10.29544708455652727911921764349

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