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Properties

Label 2-560-112.109-c1-0-35
Degree $2$
Conductor $560$
Sign $0.991 - 0.132i$
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  + (−0.0474 + 1.41i)2-s + (−1.53 + 0.412i)3-s + (−1.99 − 0.134i)4-s + (0.965 + 0.258i)5-s + (−0.509 − 2.19i)6-s + (2.59 − 0.524i)7-s + (0.284 − 2.81i)8-s + (−0.399 + 0.230i)9-s + (−0.411 + 1.35i)10-s + (−1.22 − 4.57i)11-s + (3.12 − 0.616i)12-s + (−2.24 − 2.24i)13-s + (0.618 + 3.69i)14-s − 1.59·15-s + (3.96 + 0.535i)16-s + (1.27 − 2.20i)17-s + ⋯
L(s)  = 1  + (−0.0335 + 0.999i)2-s + (−0.888 + 0.238i)3-s + (−0.997 − 0.0670i)4-s + (0.431 + 0.115i)5-s + (−0.208 − 0.896i)6-s + (0.980 − 0.198i)7-s + (0.100 − 0.994i)8-s + (−0.133 + 0.0768i)9-s + (−0.130 + 0.427i)10-s + (−0.369 − 1.37i)11-s + (0.902 − 0.177i)12-s + (−0.622 − 0.622i)13-s + (0.165 + 0.986i)14-s − 0.411·15-s + (0.991 + 0.133i)16-s + (0.308 − 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.991 - 0.132i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923264 + 0.0614911i\)
\(L(\frac12)\) \(\approx\) \(0.923264 + 0.0614911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0474 - 1.41i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-2.59 + 0.524i)T \)
good3 \( 1 + (1.53 - 0.412i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.22 + 4.57i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.24 + 2.24i)T + 13iT^{2} \)
17 \( 1 + (-1.27 + 2.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.963 + 3.59i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.29 + 1.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.51 - 3.51i)T + 29iT^{2} \)
31 \( 1 + (1.02 - 1.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.10 + 0.295i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.11iT - 41T^{2} \)
43 \( 1 + (-3.62 + 3.62i)T - 43iT^{2} \)
47 \( 1 + (1.07 + 1.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.52 + 9.43i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.72 + 6.44i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.65 - 9.92i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-14.8 + 3.99i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + (5.41 + 3.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.54 - 7.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.5 + 12.5i)T + 83iT^{2} \)
89 \( 1 + (-1.45 + 0.841i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.79444992529874614295625326202, −9.960064603165630177686700988140, −8.774613950487760006817083095622, −8.097366607008820889490931363369, −7.08958549899494156508874339211, −6.06509624731407776442198386979, −5.19869112159378094680930924085, −4.85053152877026600241829653227, −3.06157559499531740114355942344, −0.66848996443561986611302149141, 1.40702741497403255052013498558, 2.44133211274378876657973316891, 4.21909954658239267972311597970, 5.07656237875251099596401556019, 5.80940513221929030433222730290, 7.21751278532177204665369371690, 8.208173981848086665484055604385, 9.278015441874441229519109829810, 10.05277317763920385250829675232, 10.85705994025044775702595825971

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