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Properties

Label 2-560-20.3-c1-0-17
Degree $2$
Conductor $560$
Sign $-0.487 + 0.873i$
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  + (1.38 − 1.38i)3-s + (−0.674 − 2.13i)5-s + (−0.707 − 0.707i)7-s − 0.863i·9-s − 2.68i·11-s + (−2.30 − 2.30i)13-s + (−3.90 − 2.02i)15-s + (−2.61 + 2.61i)17-s + 3.54·19-s − 1.96·21-s + (2.44 − 2.44i)23-s + (−4.08 + 2.87i)25-s + (2.96 + 2.96i)27-s − 8.11i·29-s − 1.86i·31-s + ⋯
L(s)  = 1  + (0.802 − 0.802i)3-s + (−0.301 − 0.953i)5-s + (−0.267 − 0.267i)7-s − 0.287i·9-s − 0.809i·11-s + (−0.639 − 0.639i)13-s + (−1.00 − 0.522i)15-s + (−0.635 + 0.635i)17-s + 0.813·19-s − 0.428·21-s + (0.508 − 0.508i)23-s + (−0.817 + 0.575i)25-s + (0.571 + 0.571i)27-s − 1.50i·29-s − 0.334i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.487 + 0.873i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.487 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788084 - 1.34195i\)
\(L(\frac12)\) \(\approx\) \(0.788084 - 1.34195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.674 + 2.13i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.38 + 1.38i)T - 3iT^{2} \)
11 \( 1 + 2.68iT - 11T^{2} \)
13 \( 1 + (2.30 + 2.30i)T + 13iT^{2} \)
17 \( 1 + (2.61 - 2.61i)T - 17iT^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \)
29 \( 1 + 8.11iT - 29T^{2} \)
31 \( 1 + 1.86iT - 31T^{2} \)
37 \( 1 + (6.38 - 6.38i)T - 37iT^{2} \)
41 \( 1 - 0.997T + 41T^{2} \)
43 \( 1 + (-4.37 + 4.37i)T - 43iT^{2} \)
47 \( 1 + (-8.02 - 8.02i)T + 47iT^{2} \)
53 \( 1 + (4.08 + 4.08i)T + 53iT^{2} \)
59 \( 1 - 8.13T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (2.19 + 2.19i)T + 67iT^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (-5.38 - 5.38i)T + 73iT^{2} \)
79 \( 1 + 5.64T + 79T^{2} \)
83 \( 1 + (-1.60 + 1.60i)T - 83iT^{2} \)
89 \( 1 + 9.00iT - 89T^{2} \)
97 \( 1 + (8.80 - 8.80i)T - 97iT^{2} \)
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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

−10.41688323759237932507585149289, −9.359370526071741906182643482159, −8.503692727582069502279323503940, −7.952697077023458625671636694271, −7.12149880431924704629499090101, −5.90645867761546379811703639513, −4.79251244697470993911193121843, −3.54817819273463455017789464397, −2.34454680996632252992219549332, −0.820497033671846560360464970661, 2.35102258018215215821144951588, 3.27314458645871750591106203584, 4.21845986203406091642295180613, 5.33455494589257795286343033698, 6.92720674412618918985241651408, 7.24467417528143247960753041342, 8.665081642328197767164577256030, 9.392491988373470264606178607791, 9.990049961647530642108177217963, 10.91360912148065609582505899529

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