↑

Properties

Label 2-1805-95.84-c0-0-0
Degree 22
Conductor 18051805
Sign −0.671−0.740i-0.671 - 0.740i
Analytic cond. 0.9008120.900812
Root an. cond. 0.9491110.949111
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  + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s − 2·11-s + (−0.499 + 0.866i)16-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (−0.499 + 0.866i)36-s + (−1 − 1.73i)44-s − 0.999·45-s + 49-s + (1 − 1.73i)55-s + (1 + 1.73i)61-s − 0.999·64-s + (−0.499 − 0.866i)80-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s − 2·11-s + (−0.499 + 0.866i)16-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (−0.499 + 0.866i)36-s + (−1 − 1.73i)44-s − 0.999·45-s + 49-s + (1 − 1.73i)55-s + (1 + 1.73i)61-s − 0.999·64-s + (−0.499 − 0.866i)80-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((−0.671−0.740i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s) L(s)=((−0.671−0.740i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: −0.671−0.740i-0.671 - 0.740i
Analytic conductor: 0.9008120.900812
Root analytic conductor: 0.9491110.949111
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1805(654,⋅)\chi_{1805} (654, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :0), −0.671−0.740i)(2,\ 1805,\ (\ :0),\ -0.671 - 0.740i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.90517503570.9051750357
L(12)L(\frac12) ≈\approx 0.90517503570.9051750357
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)
bad5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1 1
good2 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
3 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
7 1−T2 1 - T^{2}
11 1+2T+T2 1 + 2T + T^{2}
13 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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)T2 1 + (-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

−10.10194378396398369010649243791, −8.662222920761846503929618367783, −7.902099249505173703172214692599, −7.52261717655196593223978452712, −6.89272211029900109895360237539, −5.76501086707982901293560452799, −4.77737323369318287143758728101, −3.80210848357441750356037224391, −2.77320480571584977700272445276, −2.25804372038258706175905864861, 0.63738001368318639160124004468, 1.97808180144457881941174612259, 3.15057363362349909747451420927, 4.37573129061761763166196690533, 5.19030032299635478293281023225, 5.78470530837588323895847250057, 6.84912811141498521671640861474, 7.59717800370047158160619429912, 8.343703000118279132387933579893, 9.284233658200396951443201340477

Graph of the ZZ-function along the critical line