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Properties

Label 2-560-112.109-c1-0-10
Degree $2$
Conductor $560$
Sign $0.487 - 0.872i$
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  + (−0.646 − 1.25i)2-s + (−0.291 + 0.0779i)3-s + (−1.16 + 1.62i)4-s + (0.965 + 0.258i)5-s + (0.286 + 0.315i)6-s + (1.63 + 2.07i)7-s + (2.79 + 0.415i)8-s + (−2.51 + 1.45i)9-s + (−0.298 − 1.38i)10-s + (−0.424 − 1.58i)11-s + (0.212 − 0.564i)12-s + (−1.70 − 1.70i)13-s + (1.55 − 3.40i)14-s − 0.301·15-s + (−1.28 − 3.78i)16-s + (−3.26 + 5.66i)17-s + ⋯
L(s)  = 1  + (−0.456 − 0.889i)2-s + (−0.168 + 0.0450i)3-s + (−0.582 + 0.812i)4-s + (0.431 + 0.115i)5-s + (0.116 + 0.128i)6-s + (0.618 + 0.785i)7-s + (0.989 + 0.146i)8-s + (−0.839 + 0.484i)9-s + (−0.0944 − 0.437i)10-s + (−0.128 − 0.478i)11-s + (0.0612 − 0.162i)12-s + (−0.472 − 0.472i)13-s + (0.416 − 0.909i)14-s − 0.0778·15-s + (−0.321 − 0.946i)16-s + (−0.792 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.487 - 0.872i$
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: \((2,\ 560,\ (\ :1/2),\ 0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.667094 + 0.391384i\)
\(L(\frac12)\) \(\approx\) \(0.667094 + 0.391384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.646 + 1.25i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-1.63 - 2.07i)T \)
good3 \( 1 + (0.291 - 0.0779i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.424 + 1.58i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.70 + 1.70i)T + 13iT^{2} \)
17 \( 1 + (3.26 - 5.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.35 - 5.04i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (7.22 - 4.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.60 - 6.60i)T + 29iT^{2} \)
31 \( 1 + (-3.83 + 6.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.61 + 0.968i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.51iT - 41T^{2} \)
43 \( 1 + (-7.85 + 7.85i)T - 43iT^{2} \)
47 \( 1 + (1.80 + 3.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.70 - 6.35i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.701 - 2.61i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.75 - 10.2i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.35 - 0.361i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.46iT - 71T^{2} \)
73 \( 1 + (2.21 + 1.27i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.90 - 3.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.69 - 5.69i)T + 83iT^{2} \)
89 \( 1 + (-10.7 + 6.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.611T + 97T^{2} \)
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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

−10.64708384845244882859861328135, −10.41070243362208043617715450198, −9.148741797140317907401008293794, −8.336443001949201492650817367490, −7.87684241364250073083101506101, −6.10602203909160225603262181788, −5.37064964633963616696912818157, −4.09325124229429499250855993829, −2.72028742422778170536658872489, −1.80055373799842286160433524047, 0.51364809237418413290383496609, 2.34709308317908215225328769741, 4.46820289307508711370275576698, 4.96908356466803554262421556697, 6.35275769646515690109192324716, 6.87510200561629793045266691498, 7.943898587360329997244447928981, 8.792376228237438151893838359395, 9.599527420563725332970632690888, 10.41802927064500054327325172668

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