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Properties

Label 2-560-20.3-c1-0-4
Degree $2$
Conductor $560$
Sign $0.503 - 0.863i$
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  + (0.333 − 0.333i)3-s + (−2.17 − 0.523i)5-s + (−0.707 − 0.707i)7-s + 2.77i·9-s + 3.81i·11-s + (2.82 + 2.82i)13-s + (−0.898 + 0.550i)15-s + (−2.97 + 2.97i)17-s + 5.82·19-s − 0.471·21-s + (6.07 − 6.07i)23-s + (4.45 + 2.27i)25-s + (1.92 + 1.92i)27-s + 9.62i·29-s − 3.98i·31-s + ⋯
L(s)  = 1  + (0.192 − 0.192i)3-s + (−0.972 − 0.233i)5-s + (−0.267 − 0.267i)7-s + 0.925i·9-s + 1.15i·11-s + (0.783 + 0.783i)13-s + (−0.232 + 0.142i)15-s + (−0.720 + 0.720i)17-s + 1.33·19-s − 0.102·21-s + (1.26 − 1.26i)23-s + (0.890 + 0.454i)25-s + (0.370 + 0.370i)27-s + 1.78i·29-s − 0.716i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.503 - 0.863i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.503 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00903 + 0.579693i\)
\(L(\frac12)\) \(\approx\) \(1.00903 + 0.579693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.17 + 0.523i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.333 + 0.333i)T - 3iT^{2} \)
11 \( 1 - 3.81iT - 11T^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 13iT^{2} \)
17 \( 1 + (2.97 - 2.97i)T - 17iT^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 + (-6.07 + 6.07i)T - 23iT^{2} \)
29 \( 1 - 9.62iT - 29T^{2} \)
31 \( 1 + 3.98iT - 31T^{2} \)
37 \( 1 + (-1.50 + 1.50i)T - 37iT^{2} \)
41 \( 1 + 9.04T + 41T^{2} \)
43 \( 1 + (7.59 - 7.59i)T - 43iT^{2} \)
47 \( 1 + (4.02 + 4.02i)T + 47iT^{2} \)
53 \( 1 + (-2.08 - 2.08i)T + 53iT^{2} \)
59 \( 1 - 1.30T + 59T^{2} \)
61 \( 1 + 6.60T + 61T^{2} \)
67 \( 1 + (1.76 + 1.76i)T + 67iT^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (-7.53 - 7.53i)T + 73iT^{2} \)
79 \( 1 + 4.84T + 79T^{2} \)
83 \( 1 + (-0.140 + 0.140i)T - 83iT^{2} \)
89 \( 1 + 8.64iT - 89T^{2} \)
97 \( 1 + (-7.77 + 7.77i)T - 97iT^{2} \)
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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

−11.00781376374518086544581549134, −10.11461461455565516423894844083, −8.943284827850514698675050178217, −8.301812515190941883944212853938, −7.23203732478091354204121716690, −6.75310422261496117936198778940, −5.05970159775419987334864427863, −4.34606563781672878727665644257, −3.13923578954496264873737052676, −1.56710619892905845043860586862, 0.70938163066434369768598684967, 3.28258647812758374785276464434, 3.37345549507477044040434765935, 4.98289452770041646303649608494, 6.09163188050209775492561415362, 7.03516517508047387099897908749, 8.057716811340676702202255816390, 8.837115727295295115226720632018, 9.610422902978563888211088590911, 10.75996119029934909956066952940

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