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Properties

Label 2-560-112.109-c1-0-2
Degree $2$
Conductor $560$
Sign $-0.900 + 0.434i$
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  + (0.792 + 1.17i)2-s + (−0.0529 + 0.0141i)3-s + (−0.743 + 1.85i)4-s + (−0.965 − 0.258i)5-s + (−0.0585 − 0.0507i)6-s + (−1.09 + 2.40i)7-s + (−2.76 + 0.600i)8-s + (−2.59 + 1.49i)9-s + (−0.462 − 1.33i)10-s + (−1.11 − 4.15i)11-s + (0.0130 − 0.108i)12-s + (−1.12 − 1.12i)13-s + (−3.68 + 0.623i)14-s + 0.0548·15-s + (−2.89 − 2.76i)16-s + (−0.0460 + 0.0797i)17-s + ⋯
L(s)  = 1  + (0.560 + 0.828i)2-s + (−0.0305 + 0.00819i)3-s + (−0.371 + 0.928i)4-s + (−0.431 − 0.115i)5-s + (−0.0239 − 0.0207i)6-s + (−0.414 + 0.910i)7-s + (−0.977 + 0.212i)8-s + (−0.865 + 0.499i)9-s + (−0.146 − 0.422i)10-s + (−0.335 − 1.25i)11-s + (0.00376 − 0.0314i)12-s + (−0.311 − 0.311i)13-s + (−0.986 + 0.166i)14-s + 0.0141·15-s + (−0.723 − 0.690i)16-s + (−0.0111 + 0.0193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.900 + 0.434i$
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: \((2,\ 560,\ (\ :1/2),\ -0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156691 - 0.686254i\)
\(L(\frac12)\) \(\approx\) \(0.156691 - 0.686254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.792 - 1.17i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (1.09 - 2.40i)T \)
good3 \( 1 + (0.0529 - 0.0141i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.11 + 4.15i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.12 + 1.12i)T + 13iT^{2} \)
17 \( 1 + (0.0460 - 0.0797i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.33 - 4.97i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.97 + 1.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.54 - 1.54i)T + 29iT^{2} \)
31 \( 1 + (5.28 - 9.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.20 + 2.19i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.06iT - 41T^{2} \)
43 \( 1 + (1.10 - 1.10i)T - 43iT^{2} \)
47 \( 1 + (-5.84 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.864 - 3.22i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.164 - 0.614i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.393 + 1.46i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-11.1 + 2.98i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.35iT - 71T^{2} \)
73 \( 1 + (5.63 + 3.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.48 - 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.76 + 2.76i)T + 83iT^{2} \)
89 \( 1 + (6.27 - 3.62i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.39T + 97T^{2} \)
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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

−11.40345107415893268542225803471, −10.54563865633588932436081270077, −8.996992135041857951722524609775, −8.539013243812400841894471511771, −7.78197748553374727922191963024, −6.57169869168323688090883964075, −5.63661806319594139654328516863, −5.12528652947298168835292847615, −3.55671747690633403276681171057, −2.76878702481350858780759051945, 0.31651486811768043298780118367, 2.26132550397044886016147801734, 3.45195987821934873748511906379, 4.36197638064465175583532418283, 5.32780230700358698865391257822, 6.64360898212667802905114987153, 7.33484095197110911227145513138, 8.788814373374819421436259741049, 9.615712027490777182063908741781, 10.38862689237157966674561914002

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