↑

Properties

Label 2-30e2-20.7-c1-0-40
Degree 22
Conductor 900900
Sign −0.525+0.850i-0.525 + 0.850i
Analytic cond. 7.186537.18653
Root an. cond. 2.680772.68077
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  + (1 − i)2-s − 2i·4-s + (3 − 3i)7-s + (−2 − 2i)8-s − 6i·14-s − 4·16-s + (1 + i)23-s + (−6 − 6i)28-s − 6i·29-s + (−4 + 4i)32-s − 12·41-s + (9 + 9i)43-s + 2·46-s + (7 − 7i)47-s − 11i·49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (1.13 − 1.13i)7-s + (−0.707 − 0.707i)8-s − 1.60i·14-s − 16-s + (0.208 + 0.208i)23-s + (−1.13 − 1.13i)28-s − 1.11i·29-s + (−0.707 + 0.707i)32-s − 1.87·41-s + (1.37 + 1.37i)43-s + 0.294·46-s + (1.02 − 1.02i)47-s − 1.57i·49-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((−0.525+0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+1/2) L(s)=((−0.525+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: −0.525+0.850i-0.525 + 0.850i
Analytic conductor: 7.186537.18653
Root analytic conductor: 2.680772.68077
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ900(307,⋅)\chi_{900} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :1/2), −0.525+0.850i)(2,\ 900,\ (\ :1/2),\ -0.525 + 0.850i)

Particular Values

L(1)L(1) ≈\approx 1.16867−2.09613i1.16867 - 2.09613i
L(12)L(\frac12) ≈\approx 1.16867−2.09613i1.16867 - 2.09613i
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+(−1+i)T 1 + (-1 + i)T
3 1 1
5 1 1
good7 1+(−3+3i)T−7iT2 1 + (-3 + 3i)T - 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1−13iT2 1 - 13iT^{2}
17 1+17iT2 1 + 17iT^{2}
19 1+19T2 1 + 19T^{2}
23 1+(−1−i)T+23iT2 1 + (-1 - i)T + 23iT^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+37iT2 1 + 37iT^{2}
41 1+12T+41T2 1 + 12T + 41T^{2}
43 1+(−9−9i)T+43iT2 1 + (-9 - 9i)T + 43iT^{2}
47 1+(−7+7i)T−47iT2 1 + (-7 + 7i)T - 47iT^{2}
53 1−53iT2 1 - 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1+8T+61T2 1 + 8T + 61T^{2}
67 1+(−3+3i)T−67iT2 1 + (-3 + 3i)T - 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1−73iT2 1 - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+(−11−11i)T+83iT2 1 + (-11 - 11i)T + 83iT^{2}
89 1+6iT−89T2 1 + 6iT - 89T^{2}
97 1+97iT2 1 + 97iT^{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

−10.16343299952444866717723104132, −9.195762203984399164701767355218, −8.078992412830036582370007384411, −7.24056898474944006872413298440, −6.21313502162534461243512833285, −5.10961798185405734893363398190, −4.40357451549800893699039208209, −3.54007083781514542106327202164, −2.14267014497467420032854894666, −0.966108576971304450391810362719, 1.98528519039804122427485905946, 3.12451912343664057647706000647, 4.41505097194284748975876512516, 5.21858616755767649957089447804, 5.86220419260858020584297175769, 6.95046358721728681067822128429, 7.80471716672948027902564736696, 8.642548864725372871977947213908, 9.109383217348625344383205361102, 10.59099966305573573881295562982

Graph of the ZZ-function along the critical line