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Properties

Label 2-20e2-5.4-c1-0-0
Degree 22
Conductor 400400
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 3.194013.19401
Root an. cond. 1.787181.78718
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  + 3i·3-s + 2i·7-s − 6·9-s − 11-s + 4i·13-s − 5i·17-s + 19-s − 6·21-s + 2i·23-s − 9i·27-s + 8·29-s − 10·31-s − 3i·33-s + 6i·37-s − 12·39-s + ⋯
L(s)  = 1  + 1.73i·3-s + 0.755i·7-s − 2·9-s − 0.301·11-s + 1.10i·13-s − 1.21i·17-s + 0.229·19-s − 1.30·21-s + 0.417i·23-s − 1.73i·27-s + 1.48·29-s − 1.79·31-s − 0.522i·33-s + 0.986i·37-s − 1.92·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{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: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(49,⋅)\chi_{400} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), −0.894−0.447i)(2,\ 400,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.262733+1.11295i0.262733 + 1.11295i
L(12)L(\frac12) ≈\approx 0.262733+1.11295i0.262733 + 1.11295i
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
5 1 1
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1+5iT−17T2 1 + 5iT - 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
29 1−8T+29T2 1 - 8T + 29T^{2}
31 1+10T+31T2 1 + 10T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1−4iT−47T2 1 - 4iT - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1+iT−67T2 1 + iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1−3iT−73T2 1 - 3iT - 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1−13iT−83T2 1 - 13iT - 83T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1−14iT−97T2 1 - 14iT - 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

−11.49872412557514933520958907730, −10.69825059377453912838744929628, −9.669487069599298121277040433615, −9.229689823562002455895683396539, −8.361311322866435477509894911609, −6.87854938788815571911988126649, −5.50901059185254097494787571248, −4.86836874346180885770841587027, −3.77178671360293152835645872175, −2.59587241429448041749976204684, 0.75402159991910623317391656143, 2.16011893007561993304510332556, 3.56636821323802014617499659948, 5.33768917294660560120686567234, 6.33003125406223115386590034977, 7.19887754643194255047378783871, 7.939961448700414459634292619666, 8.635798442302891806468299240118, 10.21040167473816967659040112772, 10.94222156935275510192231394945

Graph of the ZZ-function along the critical line