Properties

Label 2-1904-476.135-c0-0-2
Degree 22
Conductor 19041904
Sign 0.9470.319i0.947 - 0.319i
Analytic cond. 0.9502190.950219
Root an. cond. 0.9747920.974792
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  + (0.258 + 0.448i)3-s + (0.707 − 0.707i)7-s + (0.366 − 0.633i)9-s + (0.965 + 1.67i)11-s − 1.73·13-s + (0.5 + 0.866i)17-s + (0.5 + 0.133i)21-s + (0.707 − 1.22i)23-s + (−0.5 − 0.866i)25-s + 0.896·27-s + (0.707 + 1.22i)31-s + (−0.500 + 0.866i)33-s + (−0.448 − 0.776i)39-s − 1.00i·49-s + (−0.258 + 0.448i)51-s + ⋯
L(s)  = 1  + (0.258 + 0.448i)3-s + (0.707 − 0.707i)7-s + (0.366 − 0.633i)9-s + (0.965 + 1.67i)11-s − 1.73·13-s + (0.5 + 0.866i)17-s + (0.5 + 0.133i)21-s + (0.707 − 1.22i)23-s + (−0.5 − 0.866i)25-s + 0.896·27-s + (0.707 + 1.22i)31-s + (−0.500 + 0.866i)33-s + (−0.448 − 0.776i)39-s − 1.00i·49-s + (−0.258 + 0.448i)51-s + ⋯

Functional equation

Λ(s)=(1904s/2ΓC(s)L(s)=((0.9470.319i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1904s/2ΓC(s)L(s)=((0.9470.319i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19041904    =    247172^{4} \cdot 7 \cdot 17
Sign: 0.9470.319i0.947 - 0.319i
Analytic conductor: 0.9502190.950219
Root analytic conductor: 0.9747920.974792
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1904(1087,)\chi_{1904} (1087, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1904, ( :0), 0.9470.319i)(2,\ 1904,\ (\ :0),\ 0.947 - 0.319i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.4072307991.407230799
L(12)L(\frac12) \approx 1.4072307991.407230799
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
17 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
good3 1+(0.2580.448i)T+(0.5+0.866i)T2 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.9651.67i)T+(0.5+0.866i)T2 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2}
13 1+1.73T+T2 1 + 1.73T + T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.707+1.22i)T+(0.50.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
29 1T2 1 - T^{2}
31 1+(0.7071.22i)T+(0.5+0.866i)T2 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1T2 1 - T^{2}
43 1T2 1 - T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(0.5+0.866i)T+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
59 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 10.517T+T2 1 - 0.517T + T^{2}
73 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.9651.67i)T+(0.50.866i)T2 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2}
83 1T2 1 - T^{2}
89 1+(0.8661.5i)T+(0.50.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
97 1T2 1 - T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−9.740248162771570853240250596059, −8.700673569528244637719563459587, −7.87001746568159633302244459892, −6.95538959104009455157782318602, −6.64714787118121580465943028301, −5.03154550080646598404801402229, −4.49014095554876158102073627022, −3.88443343824757777748563736116, −2.52242672025114502018330956210, −1.38916341217935043774936476899, 1.29745561833815317456190818249, 2.42700677316046928072194148738, 3.28201020192719296336047208311, 4.60927433503944038424059727621, 5.34657057497909776470652159709, 6.09387281382190571911162767529, 7.35769986456912140334666104618, 7.62067515437959437802463736576, 8.568441108228104584382883085086, 9.286464979299899153861114562677

Graph of the ZZ-function along the critical line