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Properties

Label 2-560-35.13-c1-0-9
Degree $2$
Conductor $560$
Sign $0.502 - 0.864i$
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  + (−1.83 + 1.83i)3-s + (1.83 + 1.28i)5-s + (2.12 − 1.57i)7-s − 3.70i·9-s + 4.70·11-s + (1.83 − 1.83i)13-s + (−5.70 + i)15-s + (−0.737 − 0.737i)17-s + 4.75·19-s + (−1 + 6.77i)21-s + (−3.70 − 3.70i)23-s + (1.70 + 4.70i)25-s + (1.28 + 1.28i)27-s − 0.701i·29-s + 8.79i·31-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)3-s + (0.818 + 0.574i)5-s + (0.802 − 0.596i)7-s − 1.23i·9-s + 1.41·11-s + (0.507 − 0.507i)13-s + (−1.47 + 0.258i)15-s + (−0.178 − 0.178i)17-s + 1.09·19-s + (−0.218 + 1.47i)21-s + (−0.771 − 0.771i)23-s + (0.340 + 0.940i)25-s + (0.247 + 0.247i)27-s − 0.130i·29-s + 1.58i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21904 + 0.701481i\)
\(L(\frac12)\) \(\approx\) \(1.21904 + 0.701481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 - 1.28i)T \)
7 \( 1 + (-2.12 + 1.57i)T \)
good3 \( 1 + (1.83 - 1.83i)T - 3iT^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 + (-1.83 + 1.83i)T - 13iT^{2} \)
17 \( 1 + (0.737 + 0.737i)T + 17iT^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 + (3.70 + 3.70i)T + 23iT^{2} \)
29 \( 1 + 0.701iT - 29T^{2} \)
31 \( 1 - 8.79iT - 31T^{2} \)
37 \( 1 + (3.70 - 3.70i)T - 37iT^{2} \)
41 \( 1 - 4.75iT - 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (8.05 + 8.05i)T + 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 + 9.50iT - 61T^{2} \)
67 \( 1 + (5 - 5i)T - 67iT^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 + (-3.11 + 3.11i)T - 73iT^{2} \)
79 \( 1 + 6.70iT - 79T^{2} \)
83 \( 1 + (7.86 - 7.86i)T - 83iT^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (5.49 + 5.49i)T + 97iT^{2} \)
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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

−10.88189605760926672161704231442, −10.15159735067610326217353853592, −9.530748609555801260248143410035, −8.409835402998260099673617224597, −7.00516091643891486889316261776, −6.22465906761847689685812078326, −5.31082508796077599075887792694, −4.44123694171491415918942459113, −3.36859975231085735097403918149, −1.36884055280420249220681720359, 1.21401235187646321962082220556, 1.96113408563155824637131164384, 4.14055254073695755221140426155, 5.45708379136474874787273506745, 5.91828084217003310963638350017, 6.80214753327560676418844284650, 7.83072040574232205969532082156, 8.938504568042585640753557395016, 9.577197901792495451216349097762, 10.96633618803336524705060565697

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