↑

Properties

Label 2-1872-39.35-c0-0-0
Degree 22
Conductor 18721872
Sign −0.162−0.986i-0.162 - 0.986i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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  + 1.41i·5-s + (0.5 + 0.866i)7-s + (−1.22 − 0.707i)11-s + (0.5 + 0.866i)13-s + (1.22 + 0.707i)23-s − 1.00·25-s + (−1.22 − 0.707i)29-s − 31-s + (−1.22 + 0.707i)35-s + (0.5 + 0.866i)43-s + 1.41i·47-s + (1.00 − 1.73i)55-s + (−1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (−1.22 + 0.707i)65-s + ⋯
L(s)  = 1  + 1.41i·5-s + (0.5 + 0.866i)7-s + (−1.22 − 0.707i)11-s + (0.5 + 0.866i)13-s + (1.22 + 0.707i)23-s − 1.00·25-s + (−1.22 − 0.707i)29-s − 31-s + (−1.22 + 0.707i)35-s + (0.5 + 0.866i)43-s + 1.41i·47-s + (1.00 − 1.73i)55-s + (−1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (−1.22 + 0.707i)65-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((−0.162−0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s) L(s)=((−0.162−0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −0.162−0.986i-0.162 - 0.986i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1872(737,⋅)\chi_{1872} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :0), −0.162−0.986i)(2,\ 1872,\ (\ :0),\ -0.162 - 0.986i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0681564751.068156475
L(12)L(\frac12) ≈\approx 1.0681564751.068156475
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 1
3 1 1
13 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1−1.41iT−T2 1 - 1.41iT - T^{2}
7 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
29 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
31 1+T+T2 1 + T + T^{2}
37 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1−1.41iT−T2 1 - 1.41iT - T^{2}
53 1−T2 1 - T^{2}
59 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
73 1+T+T2 1 + T + T^{2}
79 1−T+T2 1 - T + T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)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

−9.507306878998669142621319383468, −8.960427535713105880481524403496, −7.88159349524089807796661252591, −7.41801423156554094797001985242, −6.36665264980924535256753070789, −5.76680808716137053579947968946, −4.88437031412233931285100472338, −3.56155695294615152853462184726, −2.82761206328073714035011010263, −1.91860754279898875712100839849, 0.799267715146077860608838832363, 1.98492045971916247638842383840, 3.41293786859977744830943890349, 4.46398627765299169809353965572, 5.10511010416445929708107984835, 5.66849136978678372536973133414, 7.15699950863964623707275142889, 7.61955685396489103432881329702, 8.502153526844189437723539905051, 9.007745355034732865791125510452

Graph of the ZZ-function along the critical line