↑

Properties

Label 2-3332-476.439-c0-0-1
Degree 22
Conductor 33323332
Sign 0.977+0.210i0.977 + 0.210i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
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  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−1.95 + 0.128i)5-s + (0.382 − 0.923i)8-s + (−0.793 + 0.608i)9-s + (0.128 − 1.95i)10-s + (−1 + i)13-s + (0.866 + 0.5i)16-s + (−0.608 + 0.793i)17-s + (−0.499 − 0.866i)18-s + (1.92 + 0.382i)20-s + (2.82 − 0.371i)25-s + (−0.860 − 1.12i)26-s + (−1.08 − 1.63i)29-s + (−0.608 + 0.793i)32-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−1.95 + 0.128i)5-s + (0.382 − 0.923i)8-s + (−0.793 + 0.608i)9-s + (0.128 − 1.95i)10-s + (−1 + i)13-s + (0.866 + 0.5i)16-s + (−0.608 + 0.793i)17-s + (−0.499 − 0.866i)18-s + (1.92 + 0.382i)20-s + (2.82 − 0.371i)25-s + (−0.860 − 1.12i)26-s + (−1.08 − 1.63i)29-s + (−0.608 + 0.793i)32-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.977+0.210i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.977+0.210i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.977+0.210i0.977 + 0.210i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(1391,⋅)\chi_{3332} (1391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.977+0.210i)(2,\ 3332,\ (\ :0),\ 0.977 + 0.210i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.24338299160.2433829916
L(12)L(\frac12) ≈\approx 0.24338299160.2433829916
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)
bad2 1+(0.130−0.991i)T 1 + (0.130 - 0.991i)T
7 1 1
17 1+(0.608−0.793i)T 1 + (0.608 - 0.793i)T
good3 1+(0.793−0.608i)T2 1 + (0.793 - 0.608i)T^{2}
5 1+(1.95−0.128i)T+(0.991−0.130i)T2 1 + (1.95 - 0.128i)T + (0.991 - 0.130i)T^{2}
11 1+(0.130−0.991i)T2 1 + (0.130 - 0.991i)T^{2}
13 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
19 1+(−0.965−0.258i)T2 1 + (-0.965 - 0.258i)T^{2}
23 1+(−0.793−0.608i)T2 1 + (-0.793 - 0.608i)T^{2}
29 1+(1.08+1.63i)T+(−0.382+0.923i)T2 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2}
31 1+(−0.793+0.608i)T2 1 + (-0.793 + 0.608i)T^{2}
37 1+(0.293+0.257i)T+(0.130+0.991i)T2 1 + (0.293 + 0.257i)T + (0.130 + 0.991i)T^{2}
41 1+(−0.617+0.923i)T+(−0.382−0.923i)T2 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2}
43 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
47 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
53 1+(−0.607−0.465i)T+(0.258+0.965i)T2 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2}
59 1+(0.965−0.258i)T2 1 + (0.965 - 0.258i)T^{2}
61 1+(−1.49+0.735i)T+(0.608−0.793i)T2 1 + (-1.49 + 0.735i)T + (0.608 - 0.793i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
73 1+(−0.172+0.349i)T+(−0.608−0.793i)T2 1 + (-0.172 + 0.349i)T + (-0.608 - 0.793i)T^{2}
79 1+(0.793+0.608i)T2 1 + (0.793 + 0.608i)T^{2}
83 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
89 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
97 1+(−0.324+0.216i)T+(0.382−0.923i)T2 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)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

−8.561804097382279161561649167588, −7.905310852064726178443455150062, −7.41446417555298930882076927287, −6.79992144257887720537175170017, −5.84282170873255061320412300333, −4.88039242449208098389527798715, −4.22191404972935413977719331420, −3.66228958969135815401008228742, −2.29394091253661954429630664420, −0.22147116485545379211991940325, 0.819842699255900549429978681864, 2.62453778598338894169929945115, 3.26612017678013790337823239918, 3.92360100004240393556871873787, 4.83851897787974999691799190369, 5.40338506192221491182863516561, 6.92626807436048388404239918782, 7.53503153514174717763960967800, 8.251321262513451820172794383727, 8.799582035514301676223557652427

Graph of the ZZ-function along the critical line