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Properties

Label 2-560-140.19-c1-0-16
Degree $2$
Conductor $560$
Sign $-0.0824 + 0.996i$
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  + (−2.42 + 1.40i)3-s + (2.20 − 0.358i)5-s + (0.308 − 2.62i)7-s + (2.43 − 4.22i)9-s + (−3.87 + 2.23i)11-s − 5.47·13-s + (−4.85 + 3.96i)15-s + (−0.707 − 1.22i)17-s + (3.43 − 5.95i)19-s + (2.93 + 6.81i)21-s + (−0.308 + 0.534i)23-s + (4.74 − 1.58i)25-s + 5.25i·27-s − 3.87·29-s + (−3.43 − 5.95i)31-s + ⋯
L(s)  = 1  + (−1.40 + 0.809i)3-s + (0.987 − 0.160i)5-s + (0.116 − 0.993i)7-s + (0.812 − 1.40i)9-s + (−1.16 + 0.674i)11-s − 1.51·13-s + (−1.25 + 1.02i)15-s + (−0.171 − 0.297i)17-s + (0.788 − 1.36i)19-s + (0.640 + 1.48i)21-s + (−0.0643 + 0.111i)23-s + (0.948 − 0.316i)25-s + 1.01i·27-s − 0.719·29-s + (−0.617 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0824 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0824 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.0824 + 0.996i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.0824 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366752 - 0.398351i\)
\(L(\frac12)\) \(\approx\) \(0.366752 - 0.398351i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 0.358i)T \)
7 \( 1 + (-0.308 + 2.62i)T \)
good3 \( 1 + (2.42 - 1.40i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.87 - 2.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.47T + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.43 + 5.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.308 - 0.534i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + (3.43 + 5.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.24 + 2.44i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 - 6.09T + 43T^{2} \)
47 \( 1 + (-2.12 - 1.22i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 1.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.56 + 4.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 - 6.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.96iT - 71T^{2} \)
73 \( 1 + (2.12 + 3.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.30 + 2.48i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.06iT - 83T^{2} \)
89 \( 1 + (-10.1 - 5.84i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.25T + 97T^{2} \)
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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

−10.54433672296165480320835963199, −9.821220334487623082064154335839, −9.327569350633860641734411127921, −7.46281682643558802251598487263, −6.93460649901978296136246768896, −5.47747760833730304732251565515, −5.14545581442388664197845503874, −4.24791474807808131878006748305, −2.40078795266322654643833254134, −0.35233219312841414616569990618, 1.65931992302099916236450677733, 2.79104955050146316265993950106, 5.13870761675317497710733139599, 5.46644544696649573757672422145, 6.21825592876619808748125323081, 7.23780664616356914184791497994, 8.145036539017542642539479301458, 9.443578855229204446348085079606, 10.30420929468699975649556817699, 11.01245726689771685385464586260

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