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Properties

Label 2-560-112.109-c1-0-36
Degree $2$
Conductor $560$
Sign $0.999 - 0.0112i$
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  + (−1.41 − 0.0227i)2-s + (2.02 − 0.542i)3-s + (1.99 + 0.0642i)4-s + (0.965 + 0.258i)5-s + (−2.87 + 0.721i)6-s + (2.64 − 0.0210i)7-s + (−2.82 − 0.136i)8-s + (1.20 − 0.697i)9-s + (−1.35 − 0.387i)10-s + (0.571 + 2.13i)11-s + (4.08 − 0.954i)12-s + (1.98 + 1.98i)13-s + (−3.74 − 0.0303i)14-s + 2.09·15-s + (3.99 + 0.256i)16-s + (−1.37 + 2.38i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0160i)2-s + (1.16 − 0.313i)3-s + (0.999 + 0.0321i)4-s + (0.431 + 0.115i)5-s + (−1.17 + 0.294i)6-s + (0.999 − 0.00796i)7-s + (−0.998 − 0.0481i)8-s + (0.402 − 0.232i)9-s + (−0.430 − 0.122i)10-s + (0.172 + 0.642i)11-s + (1.17 − 0.275i)12-s + (0.549 + 0.549i)13-s + (−0.999 − 0.00810i)14-s + 0.541·15-s + (0.997 + 0.0642i)16-s + (−0.333 + 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0112i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.999 - 0.0112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64401 + 0.00920838i\)
\(L(\frac12)\) \(\approx\) \(1.64401 + 0.00920838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0227i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-2.64 + 0.0210i)T \)
good3 \( 1 + (-2.02 + 0.542i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.571 - 2.13i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.98 - 1.98i)T + 13iT^{2} \)
17 \( 1 + (1.37 - 2.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 4.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.466 - 0.269i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.196 + 0.196i)T + 29iT^{2} \)
31 \( 1 + (0.165 - 0.286i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.2 + 2.73i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.587iT - 41T^{2} \)
43 \( 1 + (-3.37 + 3.37i)T - 43iT^{2} \)
47 \( 1 + (5.41 + 9.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.58 + 5.91i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.58 - 13.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.47 + 9.25i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.15 + 1.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.38iT - 71T^{2} \)
73 \( 1 + (-2.33 - 1.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0283 - 0.0490i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.82 - 6.82i)T + 83iT^{2} \)
89 \( 1 + (-12.9 + 7.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−10.65877624612774842729698397288, −9.630180979041640604737951445653, −8.805292181542987371197561839592, −8.391072509764910353982138877592, −7.37298712821057872929132831241, −6.70561441653257631423386753933, −5.29622812631830943690889901164, −3.71781586825345283082928894972, −2.31579974028954423067572116076, −1.65431291656965058176662655963, 1.40810428924437659479208410211, 2.63710602546213637449765563360, 3.68643428196622381411554410897, 5.30737100400525266301600334816, 6.36772214149277519555691440614, 7.71217931029072075899589678296, 8.263125037057370187065432428150, 8.913311673432169958805090588018, 9.655759731077097377057510932913, 10.57637184374005470826834869614

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