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Properties

Label 2-1560-1560.1403-c0-0-4
Degree $2$
Conductor $1560$
Sign $-0.973 + 0.229i$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)8-s − 9-s − 1.00·10-s − 1.00·12-s + (−0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 + i)17-s + (−0.707 − 0.707i)18-s + (−0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)8-s − 9-s − 1.00·10-s − 1.00·12-s + (−0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 + i)17-s + (−0.707 − 0.707i)18-s + (−0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175483695\)
\(L(\frac12)\) \(\approx\) \(1.175483695\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 - iT \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.23006855854908685978643745859, −9.181460371540066783284521825432, −8.035419189884655035287114731924, −7.914896106717085390732882620065, −6.67614029781394825009949343312, −5.93931269872367222149634940592, −5.01866581593170072494926261589, −4.26530475305542719394515418278, −3.36033630425602045094744731035, −2.78567125768940623919766862562, 0.74045425211659851367426791725, 1.98583808909537944933566435786, 3.03073712906181149412029151633, 4.06919068177439757629953234196, 5.06808002884782947531880488265, 5.67056732946157086730293372168, 6.89651324810535105516244453191, 7.39349888205057143726986260400, 8.466103488921207574889386950426, 9.197607796521076055857795209308

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