↑

Properties

Label 2-1035-115.84-c0-0-1
Degree 22
Conductor 10351035
Sign 0.991−0.128i0.991 - 0.128i
Analytic cond. 0.5165320.516532
Root an. cond. 0.7187010.718701
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  + (1.80 + 0.258i)2-s + (2.21 + 0.650i)4-s + (−0.654 − 0.755i)5-s + (2.16 + 0.989i)8-s + (−0.983 − 1.53i)10-s + (1.70 + 1.09i)16-s + (−1.84 + 0.540i)17-s + (0.304 − 1.03i)19-s + (−0.959 − 2.10i)20-s + (0.142 + 0.989i)23-s + (−0.142 + 0.989i)25-s + (−0.698 + 1.53i)31-s + (0.983 + 0.852i)32-s + (−3.45 + 0.496i)34-s + (0.817 − 1.78i)38-s + ⋯
L(s)  = 1  + (1.80 + 0.258i)2-s + (2.21 + 0.650i)4-s + (−0.654 − 0.755i)5-s + (2.16 + 0.989i)8-s + (−0.983 − 1.53i)10-s + (1.70 + 1.09i)16-s + (−1.84 + 0.540i)17-s + (0.304 − 1.03i)19-s + (−0.959 − 2.10i)20-s + (0.142 + 0.989i)23-s + (−0.142 + 0.989i)25-s + (−0.698 + 1.53i)31-s + (0.983 + 0.852i)32-s + (−3.45 + 0.496i)34-s + (0.817 − 1.78i)38-s + ⋯

Functional equation

Λ(s)=(1035s/2 ΓC(s) L(s)=((0.991−0.128i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1035s/2 ΓC(s) L(s)=((0.991−0.128i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10351035    =    32⋅5⋅233^{2} \cdot 5 \cdot 23
Sign: 0.991−0.128i0.991 - 0.128i
Analytic conductor: 0.5165320.516532
Root analytic conductor: 0.7187010.718701
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1035(199,⋅)\chi_{1035} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1035, ( :0), 0.991−0.128i)(2,\ 1035,\ (\ :0),\ 0.991 - 0.128i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.4913100682.491310068
L(12)L(\frac12) ≈\approx 2.4913100682.491310068
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)
bad3 1 1
5 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
23 1+(−0.142−0.989i)T 1 + (-0.142 - 0.989i)T
good2 1+(−1.80−0.258i)T+(0.959+0.281i)T2 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2}
7 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
11 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
13 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
17 1+(1.84−0.540i)T+(0.841−0.540i)T2 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2}
19 1+(−0.304+1.03i)T+(−0.841−0.540i)T2 1 + (-0.304 + 1.03i)T + (-0.841 - 0.540i)T^{2}
29 1+(0.841−0.540i)T2 1 + (0.841 - 0.540i)T^{2}
31 1+(0.698−1.53i)T+(−0.654−0.755i)T2 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2}
37 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
41 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
43 1+(−0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
47 1+1.97iT−T2 1 + 1.97iT - T^{2}
53 1+(0.239+0.153i)T+(0.415+0.909i)T2 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2}
59 1+(0.415−0.909i)T2 1 + (0.415 - 0.909i)T^{2}
61 1+(−1.80−0.822i)T+(0.654+0.755i)T2 1 + (-1.80 - 0.822i)T + (0.654 + 0.755i)T^{2}
67 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
71 1+(−0.959−0.281i)T2 1 + (-0.959 - 0.281i)T^{2}
73 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
79 1+(0.817+1.27i)T+(−0.415+0.909i)T2 1 + (0.817 + 1.27i)T + (-0.415 + 0.909i)T^{2}
83 1+(0.544−0.627i)T+(−0.142−0.989i)T2 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2}
89 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
97 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)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

−10.55224534005618095408529924757, −9.095259419937350329248844691130, −8.381232453526720123577737882518, −7.16282845570123941714011314724, −6.76984685727074212721051762151, −5.50880072714793806390971910791, −4.91388920425011754209917282449, −4.10531202873362421567130734364, −3.30544461803083701405812679942, −1.97705588049490233501847823368, 2.18157508823499889326832636675, 2.99996059074540101041788821585, 4.06796445628218171972533343776, 4.55927289572106884568221646098, 5.79173827247193279844491984599, 6.51943916388485339046318162826, 7.21117313246775457280480793604, 8.166365537236139904474465600573, 9.519929491081151138963140881684, 10.65174671496911118122501229763

Graph of the ZZ-function along the critical line