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Properties

Label 2-560-140.27-c2-0-19
Degree $2$
Conductor $560$
Sign $-0.0289 - 0.999i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.21 + 1.21i)3-s + (4.89 + 1.03i)5-s + (6.65 + 2.17i)7-s + 6.05i·9-s + 14.5i·11-s + (−9.69 − 9.69i)13-s + (−7.19 + 4.67i)15-s + (10.1 − 10.1i)17-s + 12.8i·19-s + (−10.7 + 5.43i)21-s + (16.5 + 16.5i)23-s + (22.8 + 10.1i)25-s + (−18.2 − 18.2i)27-s − 50.7i·29-s − 59.0·31-s + ⋯
L(s)  = 1  + (−0.404 + 0.404i)3-s + (0.978 + 0.207i)5-s + (0.950 + 0.310i)7-s + 0.672i·9-s + 1.32i·11-s + (−0.745 − 0.745i)13-s + (−0.479 + 0.311i)15-s + (0.598 − 0.598i)17-s + 0.677i·19-s + (−0.510 + 0.258i)21-s + (0.719 + 0.719i)23-s + (0.914 + 0.405i)25-s + (−0.676 − 0.676i)27-s − 1.75i·29-s − 1.90·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0289 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0289 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.0289 - 0.999i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ -0.0289 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.894519114\)
\(L(\frac12)\) \(\approx\) \(1.894519114\)
\(L(2)\) 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 \( 1 + (-4.89 - 1.03i)T \)
7 \( 1 + (-6.65 - 2.17i)T \)
good3 \( 1 + (1.21 - 1.21i)T - 9iT^{2} \)
11 \( 1 - 14.5iT - 121T^{2} \)
13 \( 1 + (9.69 + 9.69i)T + 169iT^{2} \)
17 \( 1 + (-10.1 + 10.1i)T - 289iT^{2} \)
19 \( 1 - 12.8iT - 361T^{2} \)
23 \( 1 + (-16.5 - 16.5i)T + 529iT^{2} \)
29 \( 1 + 50.7iT - 841T^{2} \)
31 \( 1 + 59.0T + 961T^{2} \)
37 \( 1 + (-15.7 - 15.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 33.2iT - 1.68e3T^{2} \)
43 \( 1 + (-42.7 - 42.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-3.42 - 3.42i)T + 2.20e3iT^{2} \)
53 \( 1 + (58.4 - 58.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 34.0iT - 3.48e3T^{2} \)
61 \( 1 + 36.8iT - 3.72e3T^{2} \)
67 \( 1 + (86.3 - 86.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 45.8iT - 5.04e3T^{2} \)
73 \( 1 + (-32.2 - 32.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 102.T + 6.24e3T^{2} \)
83 \( 1 + (-21.7 + 21.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 68.0T + 7.92e3T^{2} \)
97 \( 1 + (39.0 - 39.0i)T - 9.40e3iT^{2} \)
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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

−10.72921933186242976963050193385, −9.831889607275565088007149293999, −9.438469231721522033405441493837, −7.87245695940680592690421828188, −7.40757748169409900844426031562, −5.88485825913614223965241601907, −5.22249679397350020589610522873, −4.50091939473788099182108385326, −2.65875658495730907506937614370, −1.66184557328762480659830275629, 0.794888400780792414881719632668, 1.93850342247853523713355203769, 3.50090889196362049324797292386, 4.95380766838816501931772759574, 5.69118939598709018441851930141, 6.65350270087834284726994175637, 7.48455176475671826135110514973, 8.880108594477263426249342557939, 9.153149056239122044181478613325, 10.65546310348684768065559373174

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