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Properties

Label 2-3800-152.37-c0-0-6
Degree 22
Conductor 38003800
Sign −0.382+0.923i-0.382 + 0.923i
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)2-s − 1.84·3-s + (−0.707 + 0.707i)4-s + (0.707 + 1.70i)6-s + (0.923 + 0.382i)8-s + 2.41·9-s − 1.41i·11-s + (1.30 − 1.30i)12-s − 0.765·13-s − i·16-s + (−0.923 − 2.23i)18-s + i·19-s + (−1.30 + 0.541i)22-s + (−1.70 − 0.707i)24-s + (0.292 + 0.707i)26-s − 2.61·27-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s − 1.84·3-s + (−0.707 + 0.707i)4-s + (0.707 + 1.70i)6-s + (0.923 + 0.382i)8-s + 2.41·9-s − 1.41i·11-s + (1.30 − 1.30i)12-s − 0.765·13-s − i·16-s + (−0.923 − 2.23i)18-s + i·19-s + (−1.30 + 0.541i)22-s + (−1.70 − 0.707i)24-s + (0.292 + 0.707i)26-s − 2.61·27-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.382+0.923i-0.382 + 0.923i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1101,⋅)\chi_{3800} (1101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), −0.382+0.923i)(2,\ 3800,\ (\ :0),\ -0.382 + 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.41330866090.4133086609
L(12)L(\frac12) ≈\approx 0.41330866090.4133086609
L(1)L(1) 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+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
5 1 1
19 1−iT 1 - iT
good3 1+1.84T+T2 1 + 1.84T + T^{2}
7 1+T2 1 + T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1+0.765T+T2 1 + 0.765T + T^{2}
17 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1−1.84T+T2 1 - 1.84T + T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1−1.84T+T2 1 - 1.84T + T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1+0.765T+T2 1 + 0.765T + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−0.765iT−T2 1 - 0.765iT - 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

−8.504655729019005346217703513311, −7.72809254991887763441794896864, −6.95838161732376364848602245355, −5.94515421005170350072166393019, −5.55287522359733585113632507757, −4.59566372591461259796922387648, −3.94749060311967316201114333561, −2.83156144406256263021252708944, −1.50148377331116799535529941165, −0.51044433845691587461890953830, 0.869058986565000773675567928839, 2.13493783075818219627071734273, 4.12349842887505980379877002231, 4.81784781246868365437143720671, 5.11918043372341302107863729853, 6.09997269660975271167861319084, 6.63394738995838425156368406983, 7.30222363296341418958668586178, 7.72817356946768780319933578032, 9.032499894065345255728359104420

Graph of the ZZ-function along the critical line