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Properties

Label 2-2550-5.4-c1-0-32
Degree 22
Conductor 25502550
Sign −0.894+0.447i-0.894 + 0.447i
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

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + 4i·7-s + i·8-s − 9-s − 4·11-s + i·12-s − 2i·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 − 1.20·11-s + 0.288i·12-s − 0.554i·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)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2550s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25502550    =    2⋅3⋅52⋅172 \cdot 3 \cdot 5^{2} \cdot 17
Sign: −0.894+0.447i-0.894 + 0.447i
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), −0.894+0.447i)(2,\ 2550,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.96109730360.9610973036
L(12)L(\frac12) ≈\approx 0.96109730360.9610973036
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+4T+11T2 1 + 4T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+12iT−43T2 1 + 12iT - 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1+2iT−53T2 1 + 2iT - 53T^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1+14iT−73T2 1 + 14iT - 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+18iT−97T2 1 + 18iT - 97T^{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

−8.526696157553920822526586781070, −8.061682512454208525152312623585, −7.15500795569719800791717256822, −6.03206777548489003367142518978, −5.41144885246067512606594430658, −4.78564837535476125633908619600, −3.24443869805087509369657731728, −2.67367673038326662384389447723, −1.87432373695221615464050964211, −0.34941243309231677094097767329, 1.14234966438259630109544124512, 2.85764367860221137790042945007, 3.84573451300330212044988773821, 4.50783288174807522656531790605, 5.28032134863114357356395140543, 6.12268663313949988592973792494, 7.08869841627912843901530571541, 7.69102384465031280252182372834, 8.181018576289516401880431443991, 9.484350459542729175077605349481

Graph of the ZZ-function along the critical line