↑

Properties

Label 2-1638-7.4-c1-0-38
Degree 22
Conductor 16381638
Sign −0.605−0.795i-0.605 - 0.795i
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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 + (−0.499 + 0.866i)4-s + (−1 − 1.73i)5-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.999 + 1.73i)10-s + (1.5 − 2.59i)11-s − 13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + (−0.5 − 0.866i)19-s + 1.99·20-s − 3·22-s + (0.500 − 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.316 + 0.547i)10-s + (0.452 − 0.783i)11-s − 0.277·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + (−0.114 − 0.198i)19-s + 0.447·20-s − 0.639·22-s + (0.100 − 0.173i)25-s + (0.0980 + 0.169i)26-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((−0.605−0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((−0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.605−0.795i-0.605 - 0.795i
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1638(235,⋅)\chi_{1638} (235, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), −0.605−0.795i)(2,\ 1638,\ (\ :1/2),\ -0.605 - 0.795i)

Particular Values

L(1)L(1) ≈\approx 0.35198254290.3519825429
L(12)L(\frac12) ≈\approx 0.35198254290.3519825429
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 1
7 1+(2.5+0.866i)T 1 + (2.5 + 0.866i)T
13 1+T 1 + T
good5 1+(1+1.73i)T+(−2.5+4.33i)T2 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.5+2.59i)T+(−5.5−9.52i)T2 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.5+4.33i)T+(−8.5−14.7i)T2 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.5+0.866i)T+(−9.5+16.4i)T2 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1−T+29T2 1 - T + 29T^{2}
31 1+(2−3.46i)T+(−15.5−26.8i)T2 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1−1.73i)T+(−18.5+32.0i)T2 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1+10T+43T2 1 + 10T + 43T^{2}
47 1+(0.5+0.866i)T+(−23.5+40.7i)T2 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.5−2.59i)T+(−26.5−45.8i)T2 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.5+2.59i)T+(−29.5−51.0i)T2 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.5−4.33i)T+(−30.5+52.8i)T2 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.5−4.33i)T+(−33.5−58.0i)T2 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2}
71 1−T+71T2 1 - T + 71T^{2}
73 1+(6−10.3i)T+(−36.5−63.2i)T2 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3−5.19i)T+(−39.5+68.4i)T2 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2}
83 1+16T+83T2 1 + 16T + 83T^{2}
89 1+(7+12.1i)T+(−44.5+77.0i)T2 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2}
97 1−4T+97T2 1 - 4T + 97T^{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

−8.807796729808393443257251890951, −8.422914580404262706192026121184, −7.31988821201935264494538366897, −6.62591983017440734858880239501, −5.43841329984929617584811183333, −4.54349973780420823816767985945, −3.55717248041186480767429172934, −2.86002270774807233345622454742, −1.22598226868610572368503868820, −0.16674946465191759335621931487, 1.79578229148379102515830178648, 3.16616201891322676865941832381, 3.94444374604613115678851810630, 5.13708092626654107699565717982, 6.14422389392412965132372362609, 6.73559753646165973976409876730, 7.39737822914420692905312622843, 8.228579190416011302502578097491, 9.081777044823750947308039423967, 9.938101877606767922038269098005

Graph of the ZZ-function along the critical line