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Properties

Label 2-1560-1560.1349-c0-0-2
Degree $2$
Conductor $1560$
Sign $-0.872 + 0.488i$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s + (0.866 + 0.499i)6-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s − 0.999·12-s − 13-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (0.866 + 0.499i)20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s + (0.866 + 0.499i)6-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s − 0.999·12-s − 13-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (0.866 + 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ -0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06717392729\)
\(L(\frac12)\) \(\approx\) \(0.06717392729\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.359504145562308865929698797132, −8.049131326399447852394282689838, −7.71206520858472841767538415089, −7.02352342515995336653169141074, −6.24411238120174680504624130616, −5.62081852328069182155219350527, −4.50929015634977972270348151579, −2.55972255514505600877049159427, −2.11834542637602205600736471499, −0.06972353008464826317171877562, 1.63352120373808205776458497356, 3.10181688433294231286970377255, 3.94108365881081078480346490966, 5.12475843274100895489732061141, 5.57110298786454059128633441288, 6.96847999741343284157400054131, 7.78587432591920244316233945023, 8.735912928508348893925137844610, 9.160355145380193299860613608221, 10.00990441910185301538492168799

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