↑

Properties

Label 2-520-104.101-c1-0-20
Degree 22
Conductor 520520
Sign 0.995−0.0981i0.995 - 0.0981i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 − 0.584i)2-s + (1.00 − 0.582i)3-s + (1.31 + 1.50i)4-s + 5-s + (−1.63 + 0.160i)6-s + (0.786 + 0.454i)7-s + (−0.814 − 2.70i)8-s + (−0.821 + 1.42i)9-s + (−1.28 − 0.584i)10-s + (2.01 + 3.49i)11-s + (2.20 + 0.752i)12-s + (1.21 + 3.39i)13-s + (−0.747 − 1.04i)14-s + (1.00 − 0.582i)15-s + (−0.534 + 3.96i)16-s + (1.41 − 2.45i)17-s + ⋯
L(s)  = 1  + (−0.910 − 0.413i)2-s + (0.582 − 0.336i)3-s + (0.658 + 0.752i)4-s + 0.447·5-s + (−0.669 + 0.0653i)6-s + (0.297 + 0.171i)7-s + (−0.287 − 0.957i)8-s + (−0.273 + 0.474i)9-s + (−0.407 − 0.184i)10-s + (0.608 + 1.05i)11-s + (0.636 + 0.217i)12-s + (0.336 + 0.941i)13-s + (−0.199 − 0.279i)14-s + (0.260 − 0.150i)15-s + (−0.133 + 0.991i)16-s + (0.344 − 0.596i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.995−0.0981i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.995−0.0981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.995−0.0981i0.995 - 0.0981i
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), 0.995−0.0981i)(2,\ 520,\ (\ :1/2),\ 0.995 - 0.0981i)

Particular Values

L(1)L(1) ≈\approx 1.28832+0.0634096i1.28832 + 0.0634096i
L(12)L(\frac12) ≈\approx 1.28832+0.0634096i1.28832 + 0.0634096i
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+(1.28+0.584i)T 1 + (1.28 + 0.584i)T
5 1−T 1 - T
13 1+(−1.21−3.39i)T 1 + (-1.21 - 3.39i)T
good3 1+(−1.00+0.582i)T+(1.5−2.59i)T2 1 + (-1.00 + 0.582i)T + (1.5 - 2.59i)T^{2}
7 1+(−0.786−0.454i)T+(3.5+6.06i)T2 1 + (-0.786 - 0.454i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.01−3.49i)T+(−5.5+9.52i)T2 1 + (-2.01 - 3.49i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.41+2.45i)T+(−8.5−14.7i)T2 1 + (-1.41 + 2.45i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.810+1.40i)T+(−9.5−16.4i)T2 1 + (-0.810 + 1.40i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.494+0.856i)T+(−11.5+19.9i)T2 1 + (0.494 + 0.856i)T + (-11.5 + 19.9i)T^{2}
29 1+(7.87−4.54i)T+(14.5−25.1i)T2 1 + (7.87 - 4.54i)T + (14.5 - 25.1i)T^{2}
31 1−3.84iT−31T2 1 - 3.84iT - 31T^{2}
37 1+(0.538+0.931i)T+(−18.5+32.0i)T2 1 + (0.538 + 0.931i)T + (-18.5 + 32.0i)T^{2}
41 1+(−7.68+4.43i)T+(20.5−35.5i)T2 1 + (-7.68 + 4.43i)T + (20.5 - 35.5i)T^{2}
43 1+(−9.23−5.33i)T+(21.5+37.2i)T2 1 + (-9.23 - 5.33i)T + (21.5 + 37.2i)T^{2}
47 1+11.5iT−47T2 1 + 11.5iT - 47T^{2}
53 1+10.1iT−53T2 1 + 10.1iT - 53T^{2}
59 1+(−4.67+8.09i)T+(−29.5−51.0i)T2 1 + (-4.67 + 8.09i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.159−0.0918i)T+(30.5+52.8i)T2 1 + (-0.159 - 0.0918i)T + (30.5 + 52.8i)T^{2}
67 1+(−5.74−9.95i)T+(−33.5+58.0i)T2 1 + (-5.74 - 9.95i)T + (-33.5 + 58.0i)T^{2}
71 1+(−4.27−2.46i)T+(35.5+61.4i)T2 1 + (-4.27 - 2.46i)T + (35.5 + 61.4i)T^{2}
73 1+5.44iT−73T2 1 + 5.44iT - 73T^{2}
79 1+14.8T+79T2 1 + 14.8T + 79T^{2}
83 1−14.8T+83T2 1 - 14.8T + 83T^{2}
89 1+(2.39−1.38i)T+(44.5−77.0i)T2 1 + (2.39 - 1.38i)T + (44.5 - 77.0i)T^{2}
97 1+(9.80+5.66i)T+(48.5+84.0i)T2 1 + (9.80 + 5.66i)T + (48.5 + 84.0i)T^{2}
show more
show less
   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

−10.89736484737994082725460456664, −9.736187258768394320499776935261, −9.163798247384157760122386708096, −8.418741942237630982741676771178, −7.34492425798007092348027958920, −6.80006702695614874053175734675, −5.27743834900158006038107525059, −3.81882981946944870138486567506, −2.40870401918101502524462616558, −1.63386182501573039001668036273, 1.07080453804646296672384968134, 2.73414942019696714429865181044, 3.94386493210342530711859813451, 5.84958487006861920083479725292, 6.01055565669897889898866472337, 7.61513557412249130985893025874, 8.226342651561169086295388740634, 9.165276378782644931231078736891, 9.642998337597152318722140954064, 10.74858291997980612318396566449

Graph of the ZZ-function along the critical line