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Properties

Label 2-520-104.101-c1-0-22
Degree 22
Conductor 520520
Sign −0.134−0.990i-0.134 - 0.990i
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  + (0.837 + 1.13i)2-s + (0.654 − 0.377i)3-s + (−0.596 + 1.90i)4-s + 5-s + (0.979 + 0.429i)6-s + (0.518 + 0.299i)7-s + (−2.67 + 0.919i)8-s + (−1.21 + 2.10i)9-s + (0.837 + 1.13i)10-s + (0.495 + 0.858i)11-s + (0.331 + 1.47i)12-s + (1.20 + 3.39i)13-s + (0.0933 + 0.842i)14-s + (0.654 − 0.377i)15-s + (−3.28 − 2.27i)16-s + (3.48 − 6.03i)17-s + ⋯
L(s)  = 1  + (0.592 + 0.805i)2-s + (0.377 − 0.218i)3-s + (−0.298 + 0.954i)4-s + 0.447·5-s + (0.399 + 0.175i)6-s + (0.196 + 0.113i)7-s + (−0.945 + 0.325i)8-s + (−0.404 + 0.701i)9-s + (0.264 + 0.360i)10-s + (0.149 + 0.258i)11-s + (0.0955 + 0.425i)12-s + (0.333 + 0.942i)13-s + (0.0249 + 0.225i)14-s + (0.169 − 0.0975i)15-s + (−0.822 − 0.569i)16-s + (0.844 − 1.46i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((−0.134−0.990i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((−0.134−0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: −0.134−0.990i-0.134 - 0.990i
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ520(101,⋅)\chi_{520} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), −0.134−0.990i)(2,\ 520,\ (\ :1/2),\ -0.134 - 0.990i)

Particular Values

L(1)L(1) ≈\approx 1.45270+1.66278i1.45270 + 1.66278i
L(12)L(\frac12) ≈\approx 1.45270+1.66278i1.45270 + 1.66278i
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+(−0.837−1.13i)T 1 + (-0.837 - 1.13i)T
5 1−T 1 - T
13 1+(−1.20−3.39i)T 1 + (-1.20 - 3.39i)T
good3 1+(−0.654+0.377i)T+(1.5−2.59i)T2 1 + (-0.654 + 0.377i)T + (1.5 - 2.59i)T^{2}
7 1+(−0.518−0.299i)T+(3.5+6.06i)T2 1 + (-0.518 - 0.299i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.495−0.858i)T+(−5.5+9.52i)T2 1 + (-0.495 - 0.858i)T + (-5.5 + 9.52i)T^{2}
17 1+(−3.48+6.03i)T+(−8.5−14.7i)T2 1 + (-3.48 + 6.03i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.07−3.58i)T+(−9.5−16.4i)T2 1 + (2.07 - 3.58i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.57−4.45i)T+(−11.5+19.9i)T2 1 + (-2.57 - 4.45i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.70+2.13i)T+(14.5−25.1i)T2 1 + (-3.70 + 2.13i)T + (14.5 - 25.1i)T^{2}
31 1+10.2iT−31T2 1 + 10.2iT - 31T^{2}
37 1+(−3.38−5.86i)T+(−18.5+32.0i)T2 1 + (-3.38 - 5.86i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.99−1.15i)T+(20.5−35.5i)T2 1 + (1.99 - 1.15i)T + (20.5 - 35.5i)T^{2}
43 1+(8.49+4.90i)T+(21.5+37.2i)T2 1 + (8.49 + 4.90i)T + (21.5 + 37.2i)T^{2}
47 1+5.84iT−47T2 1 + 5.84iT - 47T^{2}
53 1−1.02iT−53T2 1 - 1.02iT - 53T^{2}
59 1+(−0.213+0.369i)T+(−29.5−51.0i)T2 1 + (-0.213 + 0.369i)T + (-29.5 - 51.0i)T^{2}
61 1+(−7.52−4.34i)T+(30.5+52.8i)T2 1 + (-7.52 - 4.34i)T + (30.5 + 52.8i)T^{2}
67 1+(5.26+9.12i)T+(−33.5+58.0i)T2 1 + (5.26 + 9.12i)T + (-33.5 + 58.0i)T^{2}
71 1+(−2.90−1.67i)T+(35.5+61.4i)T2 1 + (-2.90 - 1.67i)T + (35.5 + 61.4i)T^{2}
73 1+11.3iT−73T2 1 + 11.3iT - 73T^{2}
79 1+1.48T+79T2 1 + 1.48T + 79T^{2}
83 1+8.05T+83T2 1 + 8.05T + 83T^{2}
89 1+(−4.06+2.34i)T+(44.5−77.0i)T2 1 + (-4.06 + 2.34i)T + (44.5 - 77.0i)T^{2}
97 1+(−12.6−7.29i)T+(48.5+84.0i)T2 1 + (-12.6 - 7.29i)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

−11.54979825765130585681108300201, −9.994268493239625718075127883126, −9.143346112321845899136117940300, −8.224484013592221186814110495124, −7.48556525991064977572672579796, −6.53027187224930783525738117041, −5.51690882932725156534376011194, −4.68941891143661579606073574878, −3.36135918698594022279948546382, −2.11279209398078251222955923284, 1.16230718795732981486476656925, 2.79521532840701270198668978668, 3.57918571676848499655029044445, 4.79371183248212424991486254710, 5.85863557811674839332168235524, 6.61630774153944129744541248598, 8.376111293183044150612990934985, 8.891574601679425744358574564621, 10.04541025319371617172142447494, 10.59565419543725196186088904099

Graph of the ZZ-function along the critical line