↑

Properties

Label 2-2240-280.37-c0-0-0
Degree 22
Conductor 22402240
Sign −0.985−0.167i-0.985 - 0.167i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
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  + (−0.366 + 1.36i)3-s + (−0.258 − 0.965i)5-s + (−0.707 − 0.707i)7-s + (−0.866 − 0.5i)9-s + (−0.866 + 0.5i)11-s + (0.707 + 0.707i)13-s + 1.41·15-s + (−0.5 + 0.866i)19-s + (1.22 − 0.707i)21-s + (0.258 + 0.965i)23-s + (−0.866 + 0.499i)25-s − 1.41·29-s + (−0.707 − 1.22i)31-s + (−0.366 − 1.36i)33-s + (−0.500 + 0.866i)35-s + ⋯
L(s)  = 1  + (−0.366 + 1.36i)3-s + (−0.258 − 0.965i)5-s + (−0.707 − 0.707i)7-s + (−0.866 − 0.5i)9-s + (−0.866 + 0.5i)11-s + (0.707 + 0.707i)13-s + 1.41·15-s + (−0.5 + 0.866i)19-s + (1.22 − 0.707i)21-s + (0.258 + 0.965i)23-s + (−0.866 + 0.499i)25-s − 1.41·29-s + (−0.707 − 1.22i)31-s + (−0.366 − 1.36i)33-s + (−0.500 + 0.866i)35-s + ⋯

Functional equation

Λ(s)=(2240s/2 ΓC(s) L(s)=((−0.985−0.167i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2240s/2 ΓC(s) L(s)=((−0.985−0.167i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22402240    =    26⋅5⋅72^{6} \cdot 5 \cdot 7
Sign: −0.985−0.167i-0.985 - 0.167i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2240(737,⋅)\chi_{2240} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2240, ( :0), −0.985−0.167i)(2,\ 2240,\ (\ :0),\ -0.985 - 0.167i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.36983862130.3698386213
L(12)L(\frac12) ≈\approx 0.36983862130.3698386213
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 1
5 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
7 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
11 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
13 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
17 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
19 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(−0.258−0.965i)T+(−0.866+0.5i)T2 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1+(0.707+1.22i)T+(−0.5+0.866i)T2 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}
37 1+(−0.258−0.965i)T+(−0.866+0.5i)T2 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2}
41 1+T+T2 1 + T + T^{2}
43 1+iT2 1 + iT^{2}
47 1+(0.965−0.258i)T+(0.866−0.5i)T2 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}
53 1+(0.258−0.965i)T+(−0.866−0.5i)T2 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
67 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
71 1−1.41T+T2 1 - 1.41T + T^{2}
73 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{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

−9.610890910234444547234817576046, −9.144953298405462217386869556239, −8.095903120840793334319546245679, −7.41631236052104405706900084903, −6.23961695196509959965836089928, −5.45629075672285203425363949333, −4.70734941394789613967535919037, −3.96257200793958262552885570494, −3.47194881373948817634298372959, −1.69723695213501079369647787523, 0.25325610861390715420772487616, 2.00292436188368389246823944022, 2.83729252079276455456521985933, 3.60500853508852761862756964512, 5.25360907277955847660863062351, 5.92045510652989870160672112437, 6.65607606232771766480642519731, 7.08409023644641718941535108853, 8.031831468769821218372520426104, 8.563353529695794333484524921515

Graph of the ZZ-function along the critical line