↑

Properties

Label 2-1470-35.27-c1-0-22
Degree 22
Conductor 14701470
Sign 0.241+0.970i0.241 + 0.970i
Analytic cond. 11.738011.7380
Root an. cond. 3.426073.42607
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  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.49 − 1.66i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−2.23 + 0.122i)10-s − 0.520·11-s + (0.707 − 0.707i)12-s + (2.39 + 2.39i)13-s + (−2.23 + 0.122i)15-s − 1.00·16-s + (−0.110 + 0.110i)17-s + (0.707 − 0.707i)18-s + 6.73·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.667 − 0.744i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.706 + 0.0386i)10-s − 0.156·11-s + (0.204 − 0.204i)12-s + (0.663 + 0.663i)13-s + (−0.576 + 0.0315i)15-s − 0.250·16-s + (−0.0267 + 0.0267i)17-s + (0.166 − 0.166i)18-s + 1.54·19-s + ⋯

Functional equation

Λ(s)=(1470s/2 ΓC(s) L(s)=((0.241+0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1470s/2 ΓC(s+1/2) L(s)=((0.241+0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14701470    =    2⋅3⋅5⋅722 \cdot 3 \cdot 5 \cdot 7^{2}
Sign: 0.241+0.970i0.241 + 0.970i
Analytic conductor: 11.738011.7380
Root analytic conductor: 3.426073.42607
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1470(97,⋅)\chi_{1470} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1470, ( :1/2), 0.241+0.970i)(2,\ 1470,\ (\ :1/2),\ 0.241 + 0.970i)

Particular Values

L(1)L(1) ≈\approx 1.3446058791.344605879
L(12)L(\frac12) ≈\approx 1.3446058791.344605879
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1+(−1.49+1.66i)T 1 + (-1.49 + 1.66i)T
7 1 1
good11 1+0.520T+11T2 1 + 0.520T + 11T^{2}
13 1+(−2.39−2.39i)T+13iT2 1 + (-2.39 - 2.39i)T + 13iT^{2}
17 1+(0.110−0.110i)T−17iT2 1 + (0.110 - 0.110i)T - 17iT^{2}
19 1−6.73T+19T2 1 - 6.73T + 19T^{2}
23 1+(−0.802+0.802i)T−23iT2 1 + (-0.802 + 0.802i)T - 23iT^{2}
29 1+1.20iT−29T2 1 + 1.20iT - 29T^{2}
31 1−7.18iT−31T2 1 - 7.18iT - 31T^{2}
37 1+(−4.41−4.41i)T+37iT2 1 + (-4.41 - 4.41i)T + 37iT^{2}
41 1−1.23iT−41T2 1 - 1.23iT - 41T^{2}
43 1+(−6.27+6.27i)T−43iT2 1 + (-6.27 + 6.27i)T - 43iT^{2}
47 1+(−7.57+7.57i)T−47iT2 1 + (-7.57 + 7.57i)T - 47iT^{2}
53 1+(−0.550+0.550i)T−53iT2 1 + (-0.550 + 0.550i)T - 53iT^{2}
59 1+8.93T+59T2 1 + 8.93T + 59T^{2}
61 1−15.4iT−61T2 1 - 15.4iT - 61T^{2}
67 1+(10.4+10.4i)T+67iT2 1 + (10.4 + 10.4i)T + 67iT^{2}
71 1−3.05T+71T2 1 - 3.05T + 71T^{2}
73 1+(−3.22−3.22i)T+73iT2 1 + (-3.22 - 3.22i)T + 73iT^{2}
79 1+2.29iT−79T2 1 + 2.29iT - 79T^{2}
83 1+(3.43+3.43i)T+83iT2 1 + (3.43 + 3.43i)T + 83iT^{2}
89 1−16.3T+89T2 1 - 16.3T + 89T^{2}
97 1+(−9.40+9.40i)T−97iT2 1 + (-9.40 + 9.40i)T - 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

−9.227493025379984313318968702309, −8.791375601720376044743843741833, −7.81782877740661427559745128246, −6.98737369172027260602232437211, −6.03286587829133122580757393863, −5.23394164653959110980004926051, −4.27352586600873827470183411937, −2.98048873326306024690373165462, −1.76787130691627927487316840522, −0.899025128278616773380753080078, 1.04109045073573287789850090334, 2.58377513223097753871348401518, 3.63532242155744052867296199761, 4.94314883650609559124876878215, 5.84213087499922907811070005687, 6.19810416456475013561886774745, 7.38691820930490992840648735230, 7.84667680771227923452414623968, 9.220952040674975629393224403923, 9.500932756695375498729409193733

Graph of the ZZ-function along the critical line