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Properties

Label 2-560-16.13-c1-0-29
Degree $2$
Conductor $560$
Sign $0.643 + 0.765i$
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  + (0.381 + 1.36i)2-s + (−0.605 − 0.605i)3-s + (−1.70 + 1.03i)4-s + (−0.707 + 0.707i)5-s + (0.593 − 1.05i)6-s − i·7-s + (−2.06 − 1.93i)8-s − 2.26i·9-s + (−1.23 − 0.693i)10-s + (−0.812 + 0.812i)11-s + (1.66 + 0.406i)12-s + (−2.79 − 2.79i)13-s + (1.36 − 0.381i)14-s + 0.856·15-s + (1.84 − 3.54i)16-s + 4.69·17-s + ⋯
L(s)  = 1  + (0.269 + 0.963i)2-s + (−0.349 − 0.349i)3-s + (−0.854 + 0.519i)4-s + (−0.316 + 0.316i)5-s + (0.242 − 0.430i)6-s − 0.377i·7-s + (−0.730 − 0.683i)8-s − 0.755i·9-s + (−0.389 − 0.219i)10-s + (−0.244 + 0.244i)11-s + (0.480 + 0.117i)12-s + (−0.776 − 0.776i)13-s + (0.363 − 0.101i)14-s + 0.221·15-s + (0.461 − 0.887i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.643 + 0.765i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724115 - 0.337405i\)
\(L(\frac12)\) \(\approx\) \(0.724115 - 0.337405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.381 - 1.36i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (0.605 + 0.605i)T + 3iT^{2} \)
11 \( 1 + (0.812 - 0.812i)T - 11iT^{2} \)
13 \( 1 + (2.79 + 2.79i)T + 13iT^{2} \)
17 \( 1 - 4.69T + 17T^{2} \)
19 \( 1 + (1.00 + 1.00i)T + 19iT^{2} \)
23 \( 1 + 8.15iT - 23T^{2} \)
29 \( 1 + (-0.122 - 0.122i)T + 29iT^{2} \)
31 \( 1 + 2.46T + 31T^{2} \)
37 \( 1 + (-3.53 + 3.53i)T - 37iT^{2} \)
41 \( 1 + 4.84iT - 41T^{2} \)
43 \( 1 + (2.21 - 2.21i)T - 43iT^{2} \)
47 \( 1 + 3.94T + 47T^{2} \)
53 \( 1 + (-7.19 + 7.19i)T - 53iT^{2} \)
59 \( 1 + (6.54 - 6.54i)T - 59iT^{2} \)
61 \( 1 + (1.81 + 1.81i)T + 61iT^{2} \)
67 \( 1 + (0.162 + 0.162i)T + 67iT^{2} \)
71 \( 1 + 6.00iT - 71T^{2} \)
73 \( 1 - 7.09iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (7.83 + 7.83i)T + 83iT^{2} \)
89 \( 1 - 3.28iT - 89T^{2} \)
97 \( 1 - 0.453T + 97T^{2} \)
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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

−10.50511319359516317621135054758, −9.738904301902015825494983842217, −8.615180575093824826484049784392, −7.66397413083333907697105095100, −7.05784793719406064069707633053, −6.17543070283818441871598444031, −5.22893881697755431915718655688, −4.12070458014699830066484152310, −2.98589451265163655407544724780, −0.44580874989943230416913554391, 1.67779417830703602922374466059, 3.05281392516932651761463488111, 4.25325654132209162777612211136, 5.14497838833742214050041167132, 5.82151561660824562395508813753, 7.49987204870574676012341743067, 8.390935132005203462136760429544, 9.511558864138529308971935540364, 10.01568868307944367052295589450, 11.07581282196856245600235115013

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