↑

Properties

Label 2-10e2-20.19-c8-0-34
Degree 22
Conductor 100100
Sign 0.774+0.632i0.774 + 0.632i
Analytic cond. 40.737840.7378
Root an. cond. 6.382626.38262
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (12.4 − 10i)2-s − 99.9·3-s + (56 − 249. i)4-s + (−1.24e3 + 999. i)6-s + 1.39e3·7-s + (−1.79e3 − 3.68e3i)8-s + 3.42e3·9-s + 1.84e4i·11-s + (−5.59e3 + 2.49e4i)12-s + 5.47e3i·13-s + (1.74e4 − 1.39e4i)14-s + (−5.92e4 − 2.79e4i)16-s + 7.30e4i·17-s + (4.27e4 − 3.42e4i)18-s − 1.94e4i·19-s + ⋯
L(s)  = 1  + (0.780 − 0.625i)2-s − 1.23·3-s + (0.218 − 0.975i)4-s + (−0.962 + 0.770i)6-s + 0.582·7-s + (−0.439 − 0.898i)8-s + 0.521·9-s + 1.26i·11-s + (−0.269 + 1.20i)12-s + 0.191i·13-s + (0.454 − 0.364i)14-s + (−0.904 − 0.426i)16-s + 0.875i·17-s + (0.407 − 0.326i)18-s − 0.149i·19-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s) L(s)=((0.774+0.632i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+4) L(s)=((0.774+0.632i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 0.774+0.632i0.774 + 0.632i
Analytic conductor: 40.737840.7378
Root analytic conductor: 6.382626.38262
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ100(99,⋅)\chi_{100} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 100, ( :4), 0.774+0.632i)(2,\ 100,\ (\ :4),\ 0.774 + 0.632i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.90541−0.678500i1.90541 - 0.678500i
L(12)L(\frac12) ≈\approx 1.90541−0.678500i1.90541 - 0.678500i
L(5)L(5) 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+(−12.4+10i)T 1 + (-12.4 + 10i)T
5 1 1
good3 1+99.9T+6.56e3T2 1 + 99.9T + 6.56e3T^{2}
7 1−1.39e3T+5.76e6T2 1 - 1.39e3T + 5.76e6T^{2}
11 1−1.84e4iT−2.14e8T2 1 - 1.84e4iT - 2.14e8T^{2}
13 1−5.47e3iT−8.15e8T2 1 - 5.47e3iT - 8.15e8T^{2}
17 1−7.30e4iT−6.97e9T2 1 - 7.30e4iT - 6.97e9T^{2}
19 1+1.94e4iT−1.69e10T2 1 + 1.94e4iT - 1.69e10T^{2}
23 1−2.37e5T+7.83e10T2 1 - 2.37e5T + 7.83e10T^{2}
29 1−1.28e5T+5.00e11T2 1 - 1.28e5T + 5.00e11T^{2}
31 1−6.79e4iT−8.52e11T2 1 - 6.79e4iT - 8.52e11T^{2}
37 1+3.47e6iT−3.51e12T2 1 + 3.47e6iT - 3.51e12T^{2}
41 1−2.14e6T+7.98e12T2 1 - 2.14e6T + 7.98e12T^{2}
43 1−5.92e6T+1.16e13T2 1 - 5.92e6T + 1.16e13T^{2}
47 1−7.62e6T+2.38e13T2 1 - 7.62e6T + 2.38e13T^{2}
53 1+8.24e5iT−6.22e13T2 1 + 8.24e5iT - 6.22e13T^{2}
59 1+3.72e6iT−1.46e14T2 1 + 3.72e6iT - 1.46e14T^{2}
61 1+1.47e7T+1.91e14T2 1 + 1.47e7T + 1.91e14T^{2}
67 1−1.52e7T+4.06e14T2 1 - 1.52e7T + 4.06e14T^{2}
71 1−1.19e6iT−6.45e14T2 1 - 1.19e6iT - 6.45e14T^{2}
73 1−5.72e6iT−8.06e14T2 1 - 5.72e6iT - 8.06e14T^{2}
79 1−3.59e7iT−1.51e15T2 1 - 3.59e7iT - 1.51e15T^{2}
83 1−5.19e7T+2.25e15T2 1 - 5.19e7T + 2.25e15T^{2}
89 1−8.33e7T+3.93e15T2 1 - 8.33e7T + 3.93e15T^{2}
97 1−1.20e8iT−7.83e15T2 1 - 1.20e8iT - 7.83e15T^{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

−12.25225481705429538781857508678, −11.11043805322940385782350800311, −10.59782917104236340226055044393, −9.261400447648240787522565066430, −7.28130181556114174678556267424, −6.08560739930847843434080203054, −5.08320770220900098195877677616, −4.15543173806726832947544351467, −2.20001888641156286560004395033, −0.872563177910579807577212355639, 0.73561387542887190921855179860, 2.97334499042874556738589683028, 4.61221185586976884344238850829, 5.52685017118081355350090211770, 6.37017927208305022465075357903, 7.64163071005776073308637645796, 8.860597398516756354037583530501, 10.80386081314715990633288611430, 11.45832858951779295515523399615, 12.29215958328603789289334640537

Graph of the ZZ-function along the critical line