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Properties

Label 2-560-16.13-c1-0-14
Degree $2$
Conductor $560$
Sign $0.176 + 0.984i$
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  + (−1.39 + 0.200i)2-s + (−2.22 − 2.22i)3-s + (1.91 − 0.562i)4-s + (−0.707 + 0.707i)5-s + (3.55 + 2.66i)6-s − i·7-s + (−2.57 + 1.17i)8-s + 6.86i·9-s + (0.847 − 1.13i)10-s + (3.74 − 3.74i)11-s + (−5.51 − 3.01i)12-s + (3.06 + 3.06i)13-s + (0.200 + 1.39i)14-s + 3.14·15-s + (3.36 − 2.15i)16-s − 0.284·17-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (−1.28 − 1.28i)3-s + (0.959 − 0.281i)4-s + (−0.316 + 0.316i)5-s + (1.45 + 1.08i)6-s − 0.377i·7-s + (−0.909 + 0.414i)8-s + 2.28i·9-s + (0.268 − 0.357i)10-s + (1.12 − 1.12i)11-s + (−1.59 − 0.869i)12-s + (0.850 + 0.850i)13-s + (0.0537 + 0.374i)14-s + 0.811·15-s + (0.841 − 0.539i)16-s − 0.0688·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.176 + 0.984i$
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: \((2,\ 560,\ (\ :1/2),\ 0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454567 - 0.380233i\)
\(L(\frac12)\) \(\approx\) \(0.454567 - 0.380233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.200i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (2.22 + 2.22i)T + 3iT^{2} \)
11 \( 1 + (-3.74 + 3.74i)T - 11iT^{2} \)
13 \( 1 + (-3.06 - 3.06i)T + 13iT^{2} \)
17 \( 1 + 0.284T + 17T^{2} \)
19 \( 1 + (-5.03 - 5.03i)T + 19iT^{2} \)
23 \( 1 + 0.967iT - 23T^{2} \)
29 \( 1 + (1.47 + 1.47i)T + 29iT^{2} \)
31 \( 1 + 4.08T + 31T^{2} \)
37 \( 1 + (-7.15 + 7.15i)T - 37iT^{2} \)
41 \( 1 + 1.79iT - 41T^{2} \)
43 \( 1 + (-1.05 + 1.05i)T - 43iT^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 + (-4.43 + 4.43i)T - 53iT^{2} \)
59 \( 1 + (-7.54 + 7.54i)T - 59iT^{2} \)
61 \( 1 + (7.01 + 7.01i)T + 61iT^{2} \)
67 \( 1 + (-8.75 - 8.75i)T + 67iT^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 - 9.17iT - 73T^{2} \)
79 \( 1 + 1.76T + 79T^{2} \)
83 \( 1 + (9.02 + 9.02i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−10.97471072088006433461234066416, −9.746945047626243994608723334756, −8.591989755113250730433277626564, −7.72052690815164899338410716822, −6.94701419450338612919072608795, −6.25342251532180957454125171771, −5.63439061274374680399276485492, −3.68022961162576031493546300023, −1.75670662098786088618037494769, −0.74754608336827746432989974265, 1.05197014226247468938374672270, 3.27908172359770573990697980603, 4.39373777521175766075130372804, 5.45759880588831958855355430364, 6.37906298857689896735644671814, 7.35092062462719908130876261951, 8.717892159432101053568595676780, 9.448568740562462169268959799976, 9.957966066518509020289187816293, 10.98847017688585644982433579861

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