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Properties

Label 2-560-20.7-c1-0-13
Degree $2$
Conductor $560$
Sign $-0.496 + 0.868i$
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  + (−0.333 − 0.333i)3-s + (−2.17 + 0.523i)5-s + (0.707 − 0.707i)7-s − 2.77i·9-s + 3.81i·11-s + (2.82 − 2.82i)13-s + (0.898 + 0.550i)15-s + (−2.97 − 2.97i)17-s − 5.82·19-s − 0.471·21-s + (−6.07 − 6.07i)23-s + (4.45 − 2.27i)25-s + (−1.92 + 1.92i)27-s − 9.62i·29-s − 3.98i·31-s + ⋯
L(s)  = 1  + (−0.192 − 0.192i)3-s + (−0.972 + 0.233i)5-s + (0.267 − 0.267i)7-s − 0.925i·9-s + 1.15i·11-s + (0.783 − 0.783i)13-s + (0.232 + 0.142i)15-s + (−0.720 − 0.720i)17-s − 1.33·19-s − 0.102·21-s + (−1.26 − 1.26i)23-s + (0.890 − 0.454i)25-s + (−0.370 + 0.370i)27-s − 1.78i·29-s − 0.716i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.496 + 0.868i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.496 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361884 - 0.623717i\)
\(L(\frac12)\) \(\approx\) \(0.361884 - 0.623717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 0.523i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.333 + 0.333i)T + 3iT^{2} \)
11 \( 1 - 3.81iT - 11T^{2} \)
13 \( 1 + (-2.82 + 2.82i)T - 13iT^{2} \)
17 \( 1 + (2.97 + 2.97i)T + 17iT^{2} \)
19 \( 1 + 5.82T + 19T^{2} \)
23 \( 1 + (6.07 + 6.07i)T + 23iT^{2} \)
29 \( 1 + 9.62iT - 29T^{2} \)
31 \( 1 + 3.98iT - 31T^{2} \)
37 \( 1 + (-1.50 - 1.50i)T + 37iT^{2} \)
41 \( 1 + 9.04T + 41T^{2} \)
43 \( 1 + (-7.59 - 7.59i)T + 43iT^{2} \)
47 \( 1 + (-4.02 + 4.02i)T - 47iT^{2} \)
53 \( 1 + (-2.08 + 2.08i)T - 53iT^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 + 6.60T + 61T^{2} \)
67 \( 1 + (-1.76 + 1.76i)T - 67iT^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (-7.53 + 7.53i)T - 73iT^{2} \)
79 \( 1 - 4.84T + 79T^{2} \)
83 \( 1 + (0.140 + 0.140i)T + 83iT^{2} \)
89 \( 1 - 8.64iT - 89T^{2} \)
97 \( 1 + (-7.77 - 7.77i)T + 97iT^{2} \)
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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

−10.57151331006332966998924275570, −9.690133935403928246151037188948, −8.499453380908716271182845145017, −7.84091625329480830864221148628, −6.80548975256902264569068797773, −6.13299479202316149693207806353, −4.49313434949754633468652575805, −3.94795308176321587298390499383, −2.39798319620360501393075648690, −0.41815887736039334508172139685, 1.79950562638009207784844636718, 3.54001542116338002886534701717, 4.36652930926651368690215245121, 5.47074678168808942048369478040, 6.46743207653385057227796169062, 7.65466792161166536252788281466, 8.552178901548730999343305994146, 8.907758853018416399933211173640, 10.67350588470944275719818465609, 10.88952837743772931923728941614

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