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Properties

Label 2-560-16.13-c1-0-0
Degree $2$
Conductor $560$
Sign $-0.505 - 0.863i$
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  + (−1.37 + 0.323i)2-s + (−1.67 − 1.67i)3-s + (1.79 − 0.889i)4-s + (0.707 − 0.707i)5-s + (2.85 + 1.76i)6-s − i·7-s + (−2.17 + 1.80i)8-s + 2.64i·9-s + (−0.745 + 1.20i)10-s + (−4.03 + 4.03i)11-s + (−4.50 − 1.51i)12-s + (−4.75 − 4.75i)13-s + (0.323 + 1.37i)14-s − 2.37·15-s + (2.41 − 3.18i)16-s + 4.01·17-s + ⋯
L(s)  = 1  + (−0.973 + 0.228i)2-s + (−0.969 − 0.969i)3-s + (0.895 − 0.444i)4-s + (0.316 − 0.316i)5-s + (1.16 + 0.722i)6-s − 0.377i·7-s + (−0.770 + 0.637i)8-s + 0.880i·9-s + (−0.235 + 0.380i)10-s + (−1.21 + 1.21i)11-s + (−1.29 − 0.437i)12-s + (−1.31 − 1.31i)13-s + (0.0863 + 0.367i)14-s − 0.613·15-s + (0.604 − 0.796i)16-s + 0.972·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.505 - 0.863i$
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: \((2,\ 560,\ (\ :1/2),\ -0.505 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00916641 + 0.0159834i\)
\(L(\frac12)\) \(\approx\) \(0.00916641 + 0.0159834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.323i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (1.67 + 1.67i)T + 3iT^{2} \)
11 \( 1 + (4.03 - 4.03i)T - 11iT^{2} \)
13 \( 1 + (4.75 + 4.75i)T + 13iT^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 + (-1.17 - 1.17i)T + 19iT^{2} \)
23 \( 1 + 0.610iT - 23T^{2} \)
29 \( 1 + (-2.42 - 2.42i)T + 29iT^{2} \)
31 \( 1 + 8.63T + 31T^{2} \)
37 \( 1 + (1.14 - 1.14i)T - 37iT^{2} \)
41 \( 1 + 1.24iT - 41T^{2} \)
43 \( 1 + (8.70 - 8.70i)T - 43iT^{2} \)
47 \( 1 - 7.92T + 47T^{2} \)
53 \( 1 + (5.64 - 5.64i)T - 53iT^{2} \)
59 \( 1 + (1.23 - 1.23i)T - 59iT^{2} \)
61 \( 1 + (-5.71 - 5.71i)T + 61iT^{2} \)
67 \( 1 + (-2.38 - 2.38i)T + 67iT^{2} \)
71 \( 1 + 1.57iT - 71T^{2} \)
73 \( 1 - 1.85iT - 73T^{2} \)
79 \( 1 - 3.14T + 79T^{2} \)
83 \( 1 + (6.01 + 6.01i)T + 83iT^{2} \)
89 \( 1 - 18.0iT - 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−10.79057838289965250215073552952, −10.19887540067346179148824727741, −9.531395637533845176726409533168, −7.968733513374330463126468489320, −7.54391307389086826969366543309, −6.83848220557862697078706829381, −5.54390892324192348918727264150, −5.17205963060059276354749009731, −2.70274976531079769708929988364, −1.38923209058327605684762432389, 0.01617054185560752678610686943, 2.27361244502245815968867523030, 3.51794701332549074033467018747, 5.09969850986766514453122456343, 5.76496459819014524775445048568, 6.89637266600069808494664243238, 7.895906507323170057756919685192, 9.024819801725701392235793672671, 9.806138476733251129548437610852, 10.37276238438457459383990110229

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