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Properties

Label 2-520-104.69-c1-0-53
Degree 22
Conductor 520520
Sign −0.879+0.475i-0.879 + 0.475i
Analytic cond. 4.152224.15222
Root an. cond. 2.037692.03769
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 − 0.684i)2-s + (−1.48 − 0.858i)3-s + (1.06 − 1.69i)4-s + 5-s + (−2.42 − 0.0445i)6-s + (−3.27 + 1.89i)7-s + (0.155 − 2.82i)8-s + (−0.0260 − 0.0450i)9-s + (1.23 − 0.684i)10-s + (1.16 − 2.01i)11-s + (−3.03 + 1.60i)12-s + (−0.0528 − 3.60i)13-s + (−2.75 + 4.58i)14-s + (−1.48 − 0.858i)15-s + (−1.74 − 3.60i)16-s + (−3.14 − 5.44i)17-s + ⋯
L(s)  = 1  + (0.875 − 0.484i)2-s + (−0.858 − 0.495i)3-s + (0.531 − 0.847i)4-s + 0.447·5-s + (−0.991 − 0.0182i)6-s + (−1.23 + 0.714i)7-s + (0.0550 − 0.998i)8-s + (−0.00867 − 0.0150i)9-s + (0.391 − 0.216i)10-s + (0.351 − 0.608i)11-s + (−0.876 + 0.463i)12-s + (−0.0146 − 0.999i)13-s + (−0.737 + 1.22i)14-s + (−0.383 − 0.221i)15-s + (−0.435 − 0.900i)16-s + (−0.763 − 1.32i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((−0.879+0.475i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((−0.879+0.475i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: −0.879+0.475i-0.879 + 0.475i
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ520(381,⋅)\chi_{520} (381, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), −0.879+0.475i)(2,\ 520,\ (\ :1/2),\ -0.879 + 0.475i)

Particular Values

L(1)L(1) ≈\approx 0.335951−1.32676i0.335951 - 1.32676i
L(12)L(\frac12) ≈\approx 0.335951−1.32676i0.335951 - 1.32676i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−1.23+0.684i)T 1 + (-1.23 + 0.684i)T
5 1−T 1 - T
13 1+(0.0528+3.60i)T 1 + (0.0528 + 3.60i)T
good3 1+(1.48+0.858i)T+(1.5+2.59i)T2 1 + (1.48 + 0.858i)T + (1.5 + 2.59i)T^{2}
7 1+(3.27−1.89i)T+(3.5−6.06i)T2 1 + (3.27 - 1.89i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.16+2.01i)T+(−5.5−9.52i)T2 1 + (-1.16 + 2.01i)T + (-5.5 - 9.52i)T^{2}
17 1+(3.14+5.44i)T+(−8.5+14.7i)T2 1 + (3.14 + 5.44i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.665+1.15i)T+(−9.5+16.4i)T2 1 + (0.665 + 1.15i)T + (-9.5 + 16.4i)T^{2}
23 1+(2.99−5.18i)T+(−11.5−19.9i)T2 1 + (2.99 - 5.18i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.31−0.758i)T+(14.5+25.1i)T2 1 + (-1.31 - 0.758i)T + (14.5 + 25.1i)T^{2}
31 1−0.175iT−31T2 1 - 0.175iT - 31T^{2}
37 1+(−2.04+3.53i)T+(−18.5−32.0i)T2 1 + (-2.04 + 3.53i)T + (-18.5 - 32.0i)T^{2}
41 1+(−10.9−6.31i)T+(20.5+35.5i)T2 1 + (-10.9 - 6.31i)T + (20.5 + 35.5i)T^{2}
43 1+(−7.71+4.45i)T+(21.5−37.2i)T2 1 + (-7.71 + 4.45i)T + (21.5 - 37.2i)T^{2}
47 1−3.15iT−47T2 1 - 3.15iT - 47T^{2}
53 1+0.254iT−53T2 1 + 0.254iT - 53T^{2}
59 1+(5.61+9.72i)T+(−29.5+51.0i)T2 1 + (5.61 + 9.72i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.168+0.0972i)T+(30.5−52.8i)T2 1 + (-0.168 + 0.0972i)T + (30.5 - 52.8i)T^{2}
67 1+(−2.63+4.55i)T+(−33.5−58.0i)T2 1 + (-2.63 + 4.55i)T + (-33.5 - 58.0i)T^{2}
71 1+(−11.0+6.36i)T+(35.5−61.4i)T2 1 + (-11.0 + 6.36i)T + (35.5 - 61.4i)T^{2}
73 1+1.93iT−73T2 1 + 1.93iT - 73T^{2}
79 1−6.68T+79T2 1 - 6.68T + 79T^{2}
83 1+6.27T+83T2 1 + 6.27T + 83T^{2}
89 1+(9.69+5.59i)T+(44.5+77.0i)T2 1 + (9.69 + 5.59i)T + (44.5 + 77.0i)T^{2}
97 1+(10.4−6.01i)T+(48.5−84.0i)T2 1 + (10.4 - 6.01i)T + (48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.89976120060278524918975995381, −9.644004166606039300225152359478, −9.171604607368882908402569886612, −7.35983982339325874464795754120, −6.25016684834776358766671411642, −6.00033110919669946346762068037, −5.05657712360313135332348301535, −3.45099565030957852296423184331, −2.52326247952343878936483304640, −0.64568847587507841722283481101, 2.34760249769605589570304541796, 4.05590418130357855803532800119, 4.42627221044200414198726524448, 5.92458913699244833851513463988, 6.34154664420354679364991667038, 7.17283682217754247460372751901, 8.524605089221684284864119301996, 9.687109131853891705326556057920, 10.51817862387994483275626891383, 11.21360653612276568912871759411

Graph of the ZZ-function along the critical line