↑

Properties

Label 2-3267-11.10-c0-0-1
Degree 22
Conductor 32673267
Sign 0.522−0.852i0.522 - 0.852i
Analytic cond. 1.630441.63044
Root an. cond. 1.276881.27688
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  + 4-s + 1.93i·7-s + 0.517i·13-s + 16-s − 1.41i·19-s − 25-s + 1.93i·28-s + 1.73·31-s + 1.41i·43-s − 2.73·49-s + 0.517i·52-s + 1.41i·61-s + 64-s − 1.73·67-s + 0.517i·73-s + ⋯
L(s)  = 1  + 4-s + 1.93i·7-s + 0.517i·13-s + 16-s − 1.41i·19-s − 25-s + 1.93i·28-s + 1.73·31-s + 1.41i·43-s − 2.73·49-s + 0.517i·52-s + 1.41i·61-s + 64-s − 1.73·67-s + 0.517i·73-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((0.522−0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.522−0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.522−0.852i0.522 - 0.852i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(2782,⋅)\chi_{3267} (2782, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.522−0.852i)(2,\ 3267,\ (\ :0),\ 0.522 - 0.852i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6014301251.601430125
L(12)L(\frac12) ≈\approx 1.6014301251.601430125
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)
bad3 1 1
11 1 1
good2 1−T2 1 - T^{2}
5 1+T2 1 + T^{2}
7 1−1.93iT−T2 1 - 1.93iT - T^{2}
13 1−0.517iT−T2 1 - 0.517iT - T^{2}
17 1−T2 1 - T^{2}
19 1+1.41iT−T2 1 + 1.41iT - T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1−1.73T+T2 1 - 1.73T + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1−1.41iT−T2 1 - 1.41iT - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1+1.73T+T2 1 + 1.73T + T^{2}
71 1+T2 1 + T^{2}
73 1−0.517iT−T2 1 - 0.517iT - T^{2}
79 1+0.517iT−T2 1 + 0.517iT - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1−T+T2 1 - T + 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

−8.929265338401141616198084325052, −8.249450495780411953343866354730, −7.45777253098177644946224897216, −6.48780612808044916235831420806, −6.10000461034407803781073997851, −5.29381553457790105101882656502, −4.41526534156070664886655197547, −2.94873373982252916440742829324, −2.58563405145722755519026087337, −1.62842713776413164374217909733, 0.987091595782885604660525575453, 1.99624627511606155583601100987, 3.29401099533779911087651522055, 3.83425013407645239092474736472, 4.78494384024374958637999385871, 5.93224957270222222737158279360, 6.47896242437598672541613328868, 7.38100288563971429712876300100, 7.71506124318885082241445901818, 8.412970529283329585347392592814

Graph of the ZZ-function along the critical line