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Properties

Label 2-1560-13.9-c1-0-21
Degree $2$
Conductor $1560$
Sign $0.0128 + 0.999i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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  + (0.5 − 0.866i)3-s + 5-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1.63 + 2.83i)11-s + (2.5 − 2.59i)13-s + (0.5 − 0.866i)15-s + (−2.63 − 4.56i)19-s − 0.999·21-s + (4.27 − 7.40i)23-s + 25-s − 0.999·27-s + (−2.27 + 3.94i)29-s + 0.274·31-s + (1.63 + 2.83i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + 0.447·5-s + (−0.188 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.493 + 0.855i)11-s + (0.693 − 0.720i)13-s + (0.129 − 0.223i)15-s + (−0.605 − 1.04i)19-s − 0.218·21-s + (0.891 − 1.54i)23-s + 0.200·25-s − 0.192·27-s + (−0.422 + 0.731i)29-s + 0.0493·31-s + (0.285 + 0.493i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.782436722\)
\(L(\frac12)\) \(\approx\) \(1.782436722\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (-2.5 + 2.59i)T \)
good7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.63 - 2.83i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.63 + 4.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.27 + 7.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.27 - 3.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.274T + 31T^{2} \)
37 \( 1 + (-3.63 + 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.27 + 5.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.137 - 0.238i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.27T + 47T^{2} \)
53 \( 1 - 9.27T + 53T^{2} \)
59 \( 1 + (-0.274 - 0.476i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.13 + 3.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.13 - 1.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 8.27T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + (-6.63 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.41 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.998813165671571044460178041932, −8.621010851788106789204090017885, −7.46890516091460673768715124480, −6.96070544918012222249819345758, −6.07952497016127975895791974655, −5.13410643810228361426802515514, −4.17012792851454133035675836127, −2.93823598035367074995142850478, −2.12732362578530929528344508307, −0.68780073689650834684624357754, 1.47551615743428101748592828330, 2.74333738071149507649510473713, 3.60633255523269079437272511480, 4.57348439745308300746798002619, 5.74438790962042197793056864932, 6.07456118542516564663981529265, 7.31179768586817945817446621428, 8.252534710614065019028372574405, 8.866512849496538238205779556407, 9.614016375968031855275063112874

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