↑

Properties

Label 2-2550-5.4-c1-0-20
Degree 22
Conductor 25502550
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 20.361820.3618
Root an. cond. 4.512414.51241
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  + i·2-s + i·3-s − 4-s − 6-s − 4i·7-s − i·8-s − 9-s − 2·11-s − i·12-s + 6i·13-s + 4·14-s + 16-s − i·17-s − i·18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 0.603·11-s − 0.288i·12-s + 1.66i·13-s + 1.06·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s − 0.917·19-s + ⋯

Functional equation

Λ(s)=(2550s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2550s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25502550    =    2⋅3⋅52⋅172 \cdot 3 \cdot 5^{2} \cdot 17
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 20.361820.3618
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2550(2449,⋅)\chi_{2550} (2449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2550, ( :1/2), 0.447−0.894i)(2,\ 2550,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.4601834951.460183495
L(12)L(\frac12) ≈\approx 1.4601834951.460183495
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−iT 1 - iT
3 1−iT 1 - iT
5 1 1
17 1+iT 1 + iT
good7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−6iT−13T2 1 - 6iT - 13T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+5iT−23T2 1 + 5iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−10T+31T2 1 - 10T + 31T^{2}
37 1−9iT−37T2 1 - 9iT - 37T^{2}
41 1−11T+41T2 1 - 11T + 41T^{2}
43 1+10iT−43T2 1 + 10iT - 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1−11iT−53T2 1 - 11iT - 53T^{2}
59 1−15T+59T2 1 - 15T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1+14iT−67T2 1 + 14iT - 67T^{2}
71 1−11T+71T2 1 - 11T + 71T^{2}
73 1+8iT−73T2 1 + 8iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−5iT−83T2 1 - 5iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−8iT−97T2 1 - 8iT - 97T^{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

−8.979641706963142822271586399806, −8.208234869543162599507136360856, −7.48282601625325442294587195209, −6.64426613762148612579762386462, −6.22400587899111351619107430839, −4.75041175774852964809176751631, −4.50602854023142110750108078345, −3.72711382141726017625883179466, −2.40385887975412528415758913542, −0.76913164906147697318826679636, 0.74134550664436845406359186341, 2.22956103492190910402914332489, 2.64596247974211042143188680008, 3.65534335159504863052349560232, 4.98796752048631405859558859072, 5.63503650627466803185498972449, 6.18577458303303873598219905120, 7.45324991615617745713253489975, 8.292336160259142959544631054070, 8.546008060917776011503918865748

Graph of the ZZ-function along the critical line