↑

Properties

Label 2-3330-37.36-c1-0-38
Degree 22
Conductor 33303330
Sign 0.986−0.164i0.986 - 0.164i
Analytic cond. 26.590126.5901
Root an. cond. 5.156565.15656
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  + i·2-s − 4-s + i·5-s + 7-s − i·8-s − 10-s − 5·11-s + i·13-s + i·14-s + 16-s − 3i·17-s + i·19-s − i·20-s − 5i·22-s − 5i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.377·7-s − 0.353i·8-s − 0.316·10-s − 1.50·11-s + 0.277i·13-s + 0.267i·14-s + 0.250·16-s − 0.727i·17-s + 0.229i·19-s − 0.223i·20-s − 1.06i·22-s − 1.04i·23-s + ⋯

Functional equation

Λ(s)=(3330s/2 ΓC(s) L(s)=((0.986−0.164i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3330s/2 ΓC(s+1/2) L(s)=((0.986−0.164i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33303330    =    2⋅32⋅5⋅372 \cdot 3^{2} \cdot 5 \cdot 37
Sign: 0.986−0.164i0.986 - 0.164i
Analytic conductor: 26.590126.5901
Root analytic conductor: 5.156565.15656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3330(2071,⋅)\chi_{3330} (2071, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3330, ( :1/2), 0.986−0.164i)(2,\ 3330,\ (\ :1/2),\ 0.986 - 0.164i)

Particular Values

L(1)L(1) ≈\approx 1.3563376351.356337635
L(12)L(\frac12) ≈\approx 1.3563376351.356337635
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−iT 1 - iT
3 1 1
5 1−iT 1 - iT
37 1+(−6+i)T 1 + (-6 + i)T
good7 1−T+7T2 1 - T + 7T^{2}
11 1+5T+11T2 1 + 5T + 11T^{2}
13 1−iT−13T2 1 - iT - 13T^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1−iT−19T2 1 - iT - 19T^{2}
23 1+5iT−23T2 1 + 5iT - 23T^{2}
29 1+8iT−29T2 1 + 8iT - 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
41 1−8T+41T2 1 - 8T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−12T+47T2 1 - 12T + 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1+10iT−59T2 1 + 10iT - 59T^{2}
61 1+2iT−61T2 1 + 2iT - 61T^{2}
67 1+2T+67T2 1 + 2T + 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1−9T+73T2 1 - 9T + 73T^{2}
79 1−12iT−79T2 1 - 12iT - 79T^{2}
83 1−15T+83T2 1 - 15T + 83T^{2}
89 1−iT−89T2 1 - iT - 89T^{2}
97 1+16iT−97T2 1 + 16iT - 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

−8.364672554672545536394830854336, −7.84210593465563758868825123247, −7.28894248506984663678851539113, −6.37394297264298744038576319345, −5.72512554712302053034666737638, −4.85958482448872533986596414630, −4.25090127148647929878864459534, −2.98078241499529770830956438319, −2.23771745466128998460607799347, −0.51777560622550167115141344598, 0.916219283107556203928735078368, 2.04391552724843100561174529793, 2.90266843032164941989041034959, 3.86998340534345196321746900518, 4.77537439400466608574764601604, 5.39880402855016171522091088185, 6.09858532968042604629183869045, 7.60557394179398900670847159036, 7.73002553865728711709980732007, 8.777185790020573194376333798208

Graph of the ZZ-function along the critical line