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Properties

Label 2-3800-152.37-c0-0-12
Degree 22
Conductor 38003800
Sign 11
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.87·3-s + 4-s − 1.87·6-s + 1.53·7-s + 8-s + 2.53·9-s − 1.87·12-s + 0.347·13-s + 1.53·14-s + 16-s + 0.347·17-s + 2.53·18-s + 19-s − 2.87·21-s − 1.87·23-s − 1.87·24-s + 0.347·26-s − 2.87·27-s + 1.53·28-s + 0.347·29-s + 32-s + 0.347·34-s + 2.53·36-s − 37-s + 38-s − 0.652·39-s + ⋯
L(s)  = 1  + 2-s − 1.87·3-s + 4-s − 1.87·6-s + 1.53·7-s + 8-s + 2.53·9-s − 1.87·12-s + 0.347·13-s + 1.53·14-s + 16-s + 0.347·17-s + 2.53·18-s + 19-s − 2.87·21-s − 1.87·23-s − 1.87·24-s + 0.347·26-s − 2.87·27-s + 1.53·28-s + 0.347·29-s + 32-s + 0.347·34-s + 2.53·36-s − 37-s + 38-s − 0.652·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 11
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: yes
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 1)(2,\ 3800,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7690372721.769037272
L(12)L(\frac12) ≈\approx 1.7690372721.769037272
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−T 1 - T
5 1 1
19 1−T 1 - T
good3 1+1.87T+T2 1 + 1.87T + T^{2}
7 1−1.53T+T2 1 - 1.53T + T^{2}
11 1−T2 1 - T^{2}
13 1−0.347T+T2 1 - 0.347T + T^{2}
17 1−0.347T+T2 1 - 0.347T + T^{2}
23 1+1.87T+T2 1 + 1.87T + T^{2}
29 1−0.347T+T2 1 - 0.347T + T^{2}
31 1−T2 1 - T^{2}
37 1+T+T2 1 + T + T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+T+T2 1 + T + T^{2}
53 1−1.53T+T2 1 - 1.53T + T^{2}
59 1+1.87T+T2 1 + 1.87T + T^{2}
61 1−T2 1 - T^{2}
67 1−1.53T+T2 1 - 1.53T + T^{2}
71 1−T2 1 - T^{2}
73 1+1.87T+T2 1 + 1.87T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - 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

−8.351353760629594578274853851324, −7.60832019804577456202142826120, −7.02678264425828841444314941909, −6.06449188826151788193384752646, −5.66455582646833053538356700807, −4.95200572745320401436093592935, −4.47188360610106782843205787481, −3.60587576232893532899609522099, −1.95667818261153896396445420352, −1.22480060681873618762671818129, 1.22480060681873618762671818129, 1.95667818261153896396445420352, 3.60587576232893532899609522099, 4.47188360610106782843205787481, 4.95200572745320401436093592935, 5.66455582646833053538356700807, 6.06449188826151788193384752646, 7.02678264425828841444314941909, 7.60832019804577456202142826120, 8.351353760629594578274853851324

Graph of the ZZ-function along the critical line