↑

Properties

Label 2-1254-209.164-c1-0-38
Degree 22
Conductor 12541254
Sign −0.662+0.749i-0.662 + 0.749i
Analytic cond. 10.013210.0132
Root an. cond. 3.164373.16437
Motivic weight 11
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)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−2.02 − 3.51i)5-s + (0.866 + 0.499i)6-s − 4.70i·7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (2.02 − 3.51i)10-s + (3.26 + 0.586i)11-s + 0.999i·12-s + (−2.31 + 4.00i)13-s + (4.07 − 2.35i)14-s + (−3.51 − 2.02i)15-s + (−0.5 − 0.866i)16-s + (−5.05 + 2.91i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.907 − 1.57i)5-s + (0.353 + 0.204i)6-s − 1.77i·7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.641 − 1.11i)10-s + (0.984 + 0.176i)11-s + 0.288i·12-s + (−0.641 + 1.11i)13-s + (1.08 − 0.628i)14-s + (−0.907 − 0.524i)15-s + (−0.125 − 0.216i)16-s + (−1.22 + 0.707i)17-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=((−0.662+0.749i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=((−0.662+0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: −0.662+0.749i-0.662 + 0.749i
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1254(373,⋅)\chi_{1254} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), −0.662+0.749i)(2,\ 1254,\ (\ :1/2),\ -0.662 + 0.749i)

Particular Values

L(1)L(1) ≈\approx 1.2022919381.202291938
L(12)L(\frac12) ≈\approx 1.2022919381.202291938
L(32)L(\frac{3}{2}) 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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
11 1+(−3.26−0.586i)T 1 + (-3.26 - 0.586i)T
19 1+(2.45+3.60i)T 1 + (2.45 + 3.60i)T
good5 1+(2.02+3.51i)T+(−2.5+4.33i)T2 1 + (2.02 + 3.51i)T + (-2.5 + 4.33i)T^{2}
7 1+4.70iT−7T2 1 + 4.70iT - 7T^{2}
13 1+(2.31−4.00i)T+(−6.5−11.2i)T2 1 + (2.31 - 4.00i)T + (-6.5 - 11.2i)T^{2}
17 1+(5.05−2.91i)T+(8.5−14.7i)T2 1 + (5.05 - 2.91i)T + (8.5 - 14.7i)T^{2}
23 1+(1.59−2.76i)T+(−11.5−19.9i)T2 1 + (1.59 - 2.76i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.36+2.36i)T+(−14.5−25.1i)T2 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2}
31 1+8.59iT−31T2 1 + 8.59iT - 31T^{2}
37 1−1.45iT−37T2 1 - 1.45iT - 37T^{2}
41 1+(−4.75−8.23i)T+(−20.5+35.5i)T2 1 + (-4.75 - 8.23i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.64−2.10i)T+(21.5−37.2i)T2 1 + (3.64 - 2.10i)T + (21.5 - 37.2i)T^{2}
47 1+(−5.09+8.82i)T+(−23.5−40.7i)T2 1 + (-5.09 + 8.82i)T + (-23.5 - 40.7i)T^{2}
53 1+(4.14+2.39i)T+(26.5+45.8i)T2 1 + (4.14 + 2.39i)T + (26.5 + 45.8i)T^{2}
59 1+(−10.8+6.28i)T+(29.5−51.0i)T2 1 + (-10.8 + 6.28i)T + (29.5 - 51.0i)T^{2}
61 1+(3.54+2.04i)T+(30.5+52.8i)T2 1 + (3.54 + 2.04i)T + (30.5 + 52.8i)T^{2}
67 1+(5.89+3.40i)T+(33.5+58.0i)T2 1 + (5.89 + 3.40i)T + (33.5 + 58.0i)T^{2}
71 1+(8.80−5.08i)T+(35.5−61.4i)T2 1 + (8.80 - 5.08i)T + (35.5 - 61.4i)T^{2}
73 1+(1.57−0.911i)T+(36.5−63.2i)T2 1 + (1.57 - 0.911i)T + (36.5 - 63.2i)T^{2}
79 1+(−0.102−0.178i)T+(−39.5+68.4i)T2 1 + (-0.102 - 0.178i)T + (-39.5 + 68.4i)T^{2}
83 1+8.68iT−83T2 1 + 8.68iT - 83T^{2}
89 1+(−4.02−2.32i)T+(44.5+77.0i)T2 1 + (-4.02 - 2.32i)T + (44.5 + 77.0i)T^{2}
97 1+(−7.58+4.37i)T+(48.5−84.0i)T2 1 + (-7.58 + 4.37i)T + (48.5 - 84.0i)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

−9.151725229006495123365397254549, −8.463832038682161480271697382215, −7.69783388688278032203890577371, −7.05822036602162502000685555728, −6.32548011312406240252826621803, −4.60279868666591207217774979123, −4.35285494493473140131580780177, −3.78658521585198734642407581913, −1.74139754749872508797152729540, −0.41053439907209740566583223815, 2.28148750034893453448267701432, 2.82277621391978948552394795968, 3.61661027861633467378058952027, 4.65134100354700970542143126190, 5.82154217204122815878618621400, 6.61010992086905761391266725266, 7.56888102564950155701136544008, 8.635018686486374899669824983256, 9.073870347319548082720430535252, 10.26082837180118375566189100149

Graph of the ZZ-function along the critical line