↑

Properties

Label 2-3751-341.278-c0-0-7
Degree 22
Conductor 37513751
Sign 0.394−0.918i0.394 - 0.918i
Analytic cond. 1.871991.87199
Root an. cond. 1.368201.36820
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.309 − 0.951i)19-s + (−0.809 + 0.587i)31-s + (−0.809 + 0.587i)35-s + (−0.809 − 0.587i)38-s + (−0.309 + 0.951i)40-s + (−0.309 − 0.951i)41-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.309 − 0.951i)19-s + (−0.809 + 0.587i)31-s + (−0.809 + 0.587i)35-s + (−0.809 − 0.587i)38-s + (−0.309 + 0.951i)40-s + (−0.309 − 0.951i)41-s + ⋯

Functional equation

Λ(s)=(3751s/2 ΓC(s) L(s)=((0.394−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3751s/2 ΓC(s) L(s)=((0.394−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37513751    =    112⋅3111^{2} \cdot 31
Sign: 0.394−0.918i0.394 - 0.918i
Analytic conductor: 1.871991.87199
Root analytic conductor: 1.368201.36820
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3751(2665,⋅)\chi_{3751} (2665, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3751, ( :0), 0.394−0.918i)(2,\ 3751,\ (\ :0),\ 0.394 - 0.918i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8782940471.878294047
L(12)L(\frac12) ≈\approx 1.8782940471.878294047
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)
bad11 1 1
31 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
good2 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
3 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
5 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
7 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
13 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
23 1−T2 1 - T^{2}
29 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
41 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.618−1.90i)T+(−0.809+0.587i)T2 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2}
53 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
59 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
61 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
67 1−2T+T2 1 - 2T + T^{2}
71 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
73 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
79 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
83 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
89 1−T2 1 - T^{2}
97 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{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

−8.865759035976756499585804414560, −8.232320479888020984348650656612, −7.27986102353502263297957841015, −6.34407447355327976325286589067, −5.62914948964720896353937230977, −5.17014741286682485294537916042, −4.18788539677489518681182815094, −3.07806847147647289718962838373, −2.58909926273536046115443110364, −1.97998059310487040622668083219, 0.794832609025574265063446231271, 1.98825885285986777616926286892, 3.49604723019356014815660060017, 3.92376382026510653570868208890, 5.01203950845598792244310935116, 5.54267293434978380573326131759, 6.23832280175916771384638210682, 6.77078528999218754723943059867, 7.65317006439107602398218689116, 8.523811297572916222778990718099

Graph of the ZZ-function along the critical line