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Properties

Label 2-560-35.4-c1-0-16
Degree $2$
Conductor $560$
Sign $0.0667 + 0.997i$
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  + (−0.866 − 0.5i)3-s + (1.23 − 1.86i)5-s + (2.59 − 0.5i)7-s + (−1 − 1.73i)9-s + 2i·13-s + (−2 + i)15-s + (1.73 + i)17-s + (−3 − 5.19i)19-s + (−2.5 − 0.866i)21-s + (2.59 − 1.5i)23-s + (−1.96 − 4.59i)25-s + 5i·27-s − 7·29-s + (1 − 1.73i)31-s + (2.26 − 5.46i)35-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.550 − 0.834i)5-s + (0.981 − 0.188i)7-s + (−0.333 − 0.577i)9-s + 0.554i·13-s + (−0.516 + 0.258i)15-s + (0.420 + 0.242i)17-s + (−0.688 − 1.19i)19-s + (−0.545 − 0.188i)21-s + (0.541 − 0.312i)23-s + (−0.392 − 0.919i)25-s + 0.962i·27-s − 1.29·29-s + (0.179 − 0.311i)31-s + (0.383 − 0.923i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.0667 + 0.997i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0667 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994604 - 0.930290i\)
\(L(\frac12)\) \(\approx\) \(0.994604 - 0.930290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−10.82804529835135974577123406339, −9.448185683549814725405565512482, −8.925012891883552730725839860352, −7.928448139993282566575233090802, −6.83731878884823410885052450475, −5.86909949795574037915749215860, −5.03963364170359558855640154414, −4.06890747941186393082438546212, −2.20686811108613444080958468964, −0.874759360086058280991463306878, 1.80922712379544592004348110989, 3.06813631202623581915344058520, 4.54803717468686975984234749137, 5.55278886784191293285632100768, 6.13900524552533842836611194216, 7.54718744272437751376642718751, 8.125951116508358189415755353184, 9.412628450970435303537711606164, 10.29760363115353598211475544089, 10.99980264848198017034405918892

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