↑

Properties

Label 2-435-145.144-c1-0-27
Degree 22
Conductor 435435
Sign 0.747+0.664i0.747 + 0.664i
Analytic cond. 3.473493.47349
Root an. cond. 1.863731.86373
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  + 2·2-s − 3-s + 2·4-s + (2 − i)5-s − 2·6-s − 4i·7-s + 9-s + (4 − 2i)10-s + i·11-s − 2·12-s − 2i·13-s − 8i·14-s + (−2 + i)15-s − 4·16-s + 6·17-s + 2·18-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + (0.894 − 0.447i)5-s − 0.816·6-s − 1.51i·7-s + 0.333·9-s + (1.26 − 0.632i)10-s + 0.301i·11-s − 0.577·12-s − 0.554i·13-s − 2.13i·14-s + (−0.516 + 0.258i)15-s − 16-s + 1.45·17-s + 0.471·18-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=((0.747+0.664i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1/2) L(s)=((0.747+0.664i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: 0.747+0.664i0.747 + 0.664i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(289,⋅)\chi_{435} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), 0.747+0.664i)(2,\ 435,\ (\ :1/2),\ 0.747 + 0.664i)

Particular Values

L(1)L(1) ≈\approx 2.45992−0.935260i2.45992 - 0.935260i
L(12)L(\frac12) ≈\approx 2.45992−0.935260i2.45992 - 0.935260i
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)
bad3 1+T 1 + T
5 1+(−2+i)T 1 + (-2 + i)T
29 1+(2+5i)T 1 + (2 + 5i)T
good2 1−2T+2T2 1 - 2T + 2T^{2}
7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1−iT−11T2 1 - iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1−9iT−23T2 1 - 9iT - 23T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1−T+37T2 1 - T + 37T^{2}
41 1−9iT−41T2 1 - 9iT - 41T^{2}
43 1−T+43T2 1 - T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+9iT−53T2 1 + 9iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1−6iT−61T2 1 - 6iT - 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1+15T+73T2 1 + 15T + 73T^{2}
79 1+4iT−79T2 1 + 4iT - 79T^{2}
83 1−7iT−83T2 1 - 7iT - 83T^{2}
89 1+2iT−89T2 1 + 2iT - 89T^{2}
97 1+11T+97T2 1 + 11T + 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

−11.31964077809128755766758793830, −10.07614116417574186181496133968, −9.777626087947438429350619065172, −7.969692709288310492536008623657, −7.00172992126185058755452452244, −5.87383952946041732115373603645, −5.34026607346045504613598208763, −4.27497789850775790366433240614, −3.33671717360574232261915277963, −1.36945748312626220732479902083, 2.22851511724058885799394735378, 3.18473962013872326817762539604, 4.74896060628684584323173366671, 5.54892767092728099639388658978, 6.10830894406878056202091453064, 6.95425659078922914311067993793, 8.696753834975057074824724346127, 9.476660412762067745707152239415, 10.66518176202523632624171352051, 11.53404899071342143673972193440

Graph of the ZZ-function along the critical line