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Properties

Label 2-520-104.101-c1-0-16
Degree 22
Conductor 520520
Sign 0.993+0.118i0.993 + 0.118i
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  + (0.237 − 1.39i)2-s + (−0.158 + 0.0913i)3-s + (−1.88 − 0.661i)4-s − 5-s + (0.0898 + 0.242i)6-s + (2.61 + 1.50i)7-s + (−1.36 + 2.47i)8-s + (−1.48 + 2.56i)9-s + (−0.237 + 1.39i)10-s + (1.37 + 2.38i)11-s + (0.358 − 0.0677i)12-s + (2.87 + 2.17i)13-s + (2.72 − 3.28i)14-s + (0.158 − 0.0913i)15-s + (3.12 + 2.49i)16-s + (1.19 − 2.06i)17-s + ⋯
L(s)  = 1  + (0.167 − 0.985i)2-s + (−0.0913 + 0.0527i)3-s + (−0.943 − 0.330i)4-s − 0.447·5-s + (0.0366 + 0.0988i)6-s + (0.987 + 0.570i)7-s + (−0.484 + 0.875i)8-s + (−0.494 + 0.856i)9-s + (−0.0749 + 0.440i)10-s + (0.415 + 0.720i)11-s + (0.103 − 0.0195i)12-s + (0.797 + 0.603i)13-s + (0.727 − 0.877i)14-s + (0.0408 − 0.0235i)15-s + (0.781 + 0.623i)16-s + (0.289 − 0.501i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.993+0.118i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.993+0.118i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.993+0.118i0.993 + 0.118i
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), 0.993+0.118i)(2,\ 520,\ (\ :1/2),\ 0.993 + 0.118i)

Particular Values

L(1)L(1) ≈\approx 1.31959−0.0781510i1.31959 - 0.0781510i
L(12)L(\frac12) ≈\approx 1.31959−0.0781510i1.31959 - 0.0781510i
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+(−0.237+1.39i)T 1 + (-0.237 + 1.39i)T
5 1+T 1 + T
13 1+(−2.87−2.17i)T 1 + (-2.87 - 2.17i)T
good3 1+(0.158−0.0913i)T+(1.5−2.59i)T2 1 + (0.158 - 0.0913i)T + (1.5 - 2.59i)T^{2}
7 1+(−2.61−1.50i)T+(3.5+6.06i)T2 1 + (-2.61 - 1.50i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.37−2.38i)T+(−5.5+9.52i)T2 1 + (-1.37 - 2.38i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.19+2.06i)T+(−8.5−14.7i)T2 1 + (-1.19 + 2.06i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.343+0.594i)T+(−9.5−16.4i)T2 1 + (-0.343 + 0.594i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.16+2.01i)T+(−11.5+19.9i)T2 1 + (1.16 + 2.01i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.86−1.65i)T+(14.5−25.1i)T2 1 + (2.86 - 1.65i)T + (14.5 - 25.1i)T^{2}
31 1−2.49iT−31T2 1 - 2.49iT - 31T^{2}
37 1+(−3.59−6.22i)T+(−18.5+32.0i)T2 1 + (-3.59 - 6.22i)T + (-18.5 + 32.0i)T^{2}
41 1+(−8.68+5.01i)T+(20.5−35.5i)T2 1 + (-8.68 + 5.01i)T + (20.5 - 35.5i)T^{2}
43 1+(1.97+1.14i)T+(21.5+37.2i)T2 1 + (1.97 + 1.14i)T + (21.5 + 37.2i)T^{2}
47 1−8.99iT−47T2 1 - 8.99iT - 47T^{2}
53 1−6.03iT−53T2 1 - 6.03iT - 53T^{2}
59 1+(−0.982+1.70i)T+(−29.5−51.0i)T2 1 + (-0.982 + 1.70i)T + (-29.5 - 51.0i)T^{2}
61 1+(−12.4−7.21i)T+(30.5+52.8i)T2 1 + (-12.4 - 7.21i)T + (30.5 + 52.8i)T^{2}
67 1+(0.739+1.28i)T+(−33.5+58.0i)T2 1 + (0.739 + 1.28i)T + (-33.5 + 58.0i)T^{2}
71 1+(13.5+7.84i)T+(35.5+61.4i)T2 1 + (13.5 + 7.84i)T + (35.5 + 61.4i)T^{2}
73 1−4.24iT−73T2 1 - 4.24iT - 73T^{2}
79 1+5.23T+79T2 1 + 5.23T + 79T^{2}
83 1+14.6T+83T2 1 + 14.6T + 83T^{2}
89 1+(−10.0+5.81i)T+(44.5−77.0i)T2 1 + (-10.0 + 5.81i)T + (44.5 - 77.0i)T^{2}
97 1+(6.21+3.58i)T+(48.5+84.0i)T2 1 + (6.21 + 3.58i)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

−11.17662373314975107241282859716, −10.19157490418017897412217524859, −9.062464277522296923318190634739, −8.451826244022722488054280958337, −7.47350138410728617507771316830, −5.90457018776185774163657201330, −4.90562802271531819916375181964, −4.18616146067379012710960316355, −2.71750927550869708420116777200, −1.56059174048572509506572036328, 0.858764928759839820817049746159, 3.48608525380979404988167730970, 4.15301978484542968518559144335, 5.53957228331809691602870182419, 6.17737520136635939200968803590, 7.34858936186787290025511406428, 8.150023278615169965598520672247, 8.725770890127479443618458136566, 9.828408363611299247219310813864, 11.09564678919432019147933830769

Graph of the ZZ-function along the critical line