Properties

Label 2-560-16.5-c1-0-6
Degree $2$
Conductor $560$
Sign $-0.949 + 0.314i$
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  + (0.517 + 1.31i)2-s + (−0.296 + 0.296i)3-s + (−1.46 + 1.36i)4-s + (−0.707 − 0.707i)5-s + (−0.543 − 0.237i)6-s + i·7-s + (−2.55 − 1.22i)8-s + 2.82i·9-s + (0.564 − 1.29i)10-s + (1.34 + 1.34i)11-s + (0.0305 − 0.838i)12-s + (−0.680 + 0.680i)13-s + (−1.31 + 0.517i)14-s + 0.419·15-s + (0.291 − 3.98i)16-s − 6.14·17-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (−0.171 + 0.171i)3-s + (−0.732 + 0.680i)4-s + (−0.316 − 0.316i)5-s + (−0.222 − 0.0967i)6-s + 0.377i·7-s + (−0.901 − 0.432i)8-s + 0.941i·9-s + (0.178 − 0.409i)10-s + (0.405 + 0.405i)11-s + (0.00882 − 0.242i)12-s + (−0.188 + 0.188i)13-s + (−0.351 + 0.138i)14-s + 0.108·15-s + (0.0728 − 0.997i)16-s − 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.949 + 0.314i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133856 - 0.829978i\)
\(L(\frac12)\) \(\approx\) \(0.133856 - 0.829978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.517 - 1.31i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (0.296 - 0.296i)T - 3iT^{2} \)
11 \( 1 + (-1.34 - 1.34i)T + 11iT^{2} \)
13 \( 1 + (0.680 - 0.680i)T - 13iT^{2} \)
17 \( 1 + 6.14T + 17T^{2} \)
19 \( 1 + (3.27 - 3.27i)T - 19iT^{2} \)
23 \( 1 + 7.61iT - 23T^{2} \)
29 \( 1 + (2.41 - 2.41i)T - 29iT^{2} \)
31 \( 1 + 9.71T + 31T^{2} \)
37 \( 1 + (-8.06 - 8.06i)T + 37iT^{2} \)
41 \( 1 - 2.60iT - 41T^{2} \)
43 \( 1 + (1.00 + 1.00i)T + 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (-6.93 - 6.93i)T + 53iT^{2} \)
59 \( 1 + (-2.09 - 2.09i)T + 59iT^{2} \)
61 \( 1 + (2.63 - 2.63i)T - 61iT^{2} \)
67 \( 1 + (4.31 - 4.31i)T - 67iT^{2} \)
71 \( 1 - 2.00iT - 71T^{2} \)
73 \( 1 - 0.827iT - 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + (2.03 - 2.03i)T - 83iT^{2} \)
89 \( 1 - 4.46iT - 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−11.31065535655599528029829284975, −10.36386354259903137898597286407, −9.114963140608892238460080606573, −8.570145873290854377589246458554, −7.61072392263089663500326600251, −6.69631868609188229262972066040, −5.75064418714188684634874812350, −4.66086607492189085352027556979, −4.12837638163654126218087841278, −2.35556065839388454573186216014, 0.42211969877561001252335028130, 2.13493806739217240747095718368, 3.55708846405305069859554702803, 4.19153360623029602142326762831, 5.56980260790964801219032278535, 6.50652007593158353276281918004, 7.49097209379081427726214351713, 9.004823567296882274057765455109, 9.310197009703986842524213550810, 10.63288717805438092973427687229

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