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Properties

Label 2-560-112.59-c1-0-35
Degree $2$
Conductor $560$
Sign $0.810 + 0.585i$
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  + (−1.17 − 0.791i)2-s + (0.282 + 1.05i)3-s + (0.746 + 1.85i)4-s + (0.965 + 0.258i)5-s + (0.504 − 1.46i)6-s + (2.60 − 0.486i)7-s + (0.594 − 2.76i)8-s + (1.56 − 0.903i)9-s + (−0.926 − 1.06i)10-s + (−1.45 − 5.42i)11-s + (−1.74 + 1.31i)12-s + (−0.831 − 0.831i)13-s + (−3.43 − 1.48i)14-s + 1.09i·15-s + (−2.88 + 2.76i)16-s + (−2.08 − 1.20i)17-s + ⋯
L(s)  = 1  + (−0.828 − 0.559i)2-s + (0.163 + 0.609i)3-s + (0.373 + 0.927i)4-s + (0.431 + 0.115i)5-s + (0.205 − 0.596i)6-s + (0.982 − 0.184i)7-s + (0.210 − 0.977i)8-s + (0.521 − 0.301i)9-s + (−0.293 − 0.337i)10-s + (−0.437 − 1.63i)11-s + (−0.504 + 0.378i)12-s + (−0.230 − 0.230i)13-s + (−0.917 − 0.397i)14-s + 0.282i·15-s + (−0.721 + 0.692i)16-s + (−0.505 − 0.291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.810 + 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19946 - 0.387635i\)
\(L(\frac12)\) \(\approx\) \(1.19946 - 0.387635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.791i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-2.60 + 0.486i)T \)
good3 \( 1 + (-0.282 - 1.05i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.45 + 5.42i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.831 + 0.831i)T + 13iT^{2} \)
17 \( 1 + (2.08 + 1.20i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.60 - 1.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.18 + 2.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.13 - 1.13i)T - 29iT^{2} \)
31 \( 1 + (-4.50 + 7.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.616 - 2.30i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.99T + 41T^{2} \)
43 \( 1 + (2.42 - 2.42i)T - 43iT^{2} \)
47 \( 1 + (-5.17 - 8.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.43 - 0.652i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-8.66 + 2.32i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.52 - 5.70i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (15.4 - 4.15i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.12T + 71T^{2} \)
73 \( 1 + (3.70 - 6.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.5 + 7.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.287 - 0.287i)T - 83iT^{2} \)
89 \( 1 + (0.657 + 1.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.9iT - 97T^{2} \)
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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

−10.56777166492691557249771911331, −9.897175736231472513547150673582, −9.020900699247266354198086309403, −8.234160041112963298125525083521, −7.43064335826307051128618473683, −6.16699248127418992481882101884, −4.88571226647444403974380361308, −3.71409875578613219949216287855, −2.64789429207705077941708187909, −1.06951022249900179449822773962, 1.55608236139149647155527750435, 2.21612600825763516175174188820, 4.69055830716668473555497600231, 5.29210032894877284559122241769, 6.74836046952484212643161126504, 7.31993480291868232430905311108, 8.053629125679413744214012833779, 8.981885517575689694455272446707, 9.952999479618593759958778877584, 10.49565610416685973144092741425

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