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Properties

Label 2-560-112.109-c1-0-14
Degree $2$
Conductor $560$
Sign $-0.995 - 0.0908i$
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  + (−0.567 + 1.29i)2-s + (−3.31 + 0.888i)3-s + (−1.35 − 1.47i)4-s + (0.965 + 0.258i)5-s + (0.732 − 4.79i)6-s + (−0.202 + 2.63i)7-s + (2.67 − 0.919i)8-s + (7.60 − 4.39i)9-s + (−0.883 + 1.10i)10-s + (0.333 + 1.24i)11-s + (5.80 + 3.67i)12-s + (4.12 + 4.12i)13-s + (−3.30 − 1.75i)14-s − 3.43·15-s + (−0.327 + 3.98i)16-s + (−0.243 + 0.422i)17-s + ⋯
L(s)  = 1  + (−0.401 + 0.915i)2-s + (−1.91 + 0.513i)3-s + (−0.677 − 0.735i)4-s + (0.431 + 0.115i)5-s + (0.298 − 1.95i)6-s + (−0.0764 + 0.997i)7-s + (0.945 − 0.325i)8-s + (2.53 − 1.46i)9-s + (−0.279 + 0.349i)10-s + (0.100 + 0.375i)11-s + (1.67 + 1.06i)12-s + (1.14 + 1.14i)13-s + (−0.882 − 0.470i)14-s − 0.886·15-s + (−0.0818 + 0.996i)16-s + (−0.0591 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.995 - 0.0908i$
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: \((2,\ 560,\ (\ :1/2),\ -0.995 - 0.0908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0262474 + 0.576834i\)
\(L(\frac12)\) \(\approx\) \(0.0262474 + 0.576834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.567 - 1.29i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (0.202 - 2.63i)T \)
good3 \( 1 + (3.31 - 0.888i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.333 - 1.24i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.12 - 4.12i)T + 13iT^{2} \)
17 \( 1 + (0.243 - 0.422i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.379 - 1.41i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.02 + 2.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0544 - 0.0544i)T + 29iT^{2} \)
31 \( 1 + (-2.35 + 4.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.26 + 2.21i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.37iT - 41T^{2} \)
43 \( 1 + (-1.70 + 1.70i)T - 43iT^{2} \)
47 \( 1 + (-2.19 - 3.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.21 - 8.28i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.25 - 4.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.870 - 3.24i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (10.6 - 2.84i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.91iT - 71T^{2} \)
73 \( 1 + (10.4 + 6.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.00 - 6.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.22 - 5.22i)T + 83iT^{2} \)
89 \( 1 + (3.29 - 1.90i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.545T + 97T^{2} \)
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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

−11.01357809930046012869032368815, −10.34173341161351505812348220635, −9.400493433464159644326720212454, −8.828565025268304275889863768305, −7.18978321478301176187325998150, −6.31681101941691560646589748455, −5.96737346553027185892380977627, −5.00097078128071668519159096654, −4.16753973288303112462794629122, −1.40845748785095634446033131653, 0.58512634916186686636758891153, 1.46645393731616046836694453379, 3.51964698128813683527643604893, 4.79688156726245185026188195891, 5.59291875060949937269502941205, 6.71753075127058120014412857504, 7.49694426883301857750490740493, 8.668314012730912528160806656222, 10.04458755243718254665532683713, 10.57854082605909072022566208900

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