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Properties

Label 2-570-57.56-c1-0-9
Degree 22
Conductor 570570
Sign 0.927+0.374i0.927 + 0.374i
Analytic cond. 4.551474.55147
Root an. cond. 2.133412.13341
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1 + 1.41i)3-s + 4-s − i·5-s + (1 − 1.41i)6-s + 3.41·7-s − 8-s + (−1.00 − 2.82i)9-s + i·10-s + 2.58i·11-s + (−1 + 1.41i)12-s − 6.24i·13-s − 3.41·14-s + (1.41 + i)15-s + 16-s − 2.82i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.577 + 0.816i)3-s + 0.5·4-s − 0.447i·5-s + (0.408 − 0.577i)6-s + 1.29·7-s − 0.353·8-s + (−0.333 − 0.942i)9-s + 0.316i·10-s + 0.779i·11-s + (−0.288 + 0.408i)12-s − 1.73i·13-s − 0.912·14-s + (0.365 + 0.258i)15-s + 0.250·16-s − 0.685i·17-s + ⋯

Functional equation

Λ(s)=(570s/2 ΓC(s) L(s)=((0.927+0.374i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(570s/2 ΓC(s+1/2) L(s)=((0.927+0.374i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 570570    =    2⋅3⋅5⋅192 \cdot 3 \cdot 5 \cdot 19
Sign: 0.927+0.374i0.927 + 0.374i
Analytic conductor: 4.551474.55147
Root analytic conductor: 2.133412.13341
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ570(341,⋅)\chi_{570} (341, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 570, ( :1/2), 0.927+0.374i)(2,\ 570,\ (\ :1/2),\ 0.927 + 0.374i)

Particular Values

L(1)L(1) ≈\approx 0.903618−0.175659i0.903618 - 0.175659i
L(12)L(\frac12) ≈\approx 0.903618−0.175659i0.903618 - 0.175659i
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+T 1 + T
3 1+(1−1.41i)T 1 + (1 - 1.41i)T
5 1+iT 1 + iT
19 1+(4.24−i)T 1 + (4.24 - i)T
good7 1−3.41T+7T2 1 - 3.41T + 7T^{2}
11 1−2.58iT−11T2 1 - 2.58iT - 11T^{2}
13 1+6.24iT−13T2 1 + 6.24iT - 13T^{2}
17 1+2.82iT−17T2 1 + 2.82iT - 17T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−9.07T+29T2 1 - 9.07T + 29T^{2}
31 1−1.17iT−31T2 1 - 1.17iT - 31T^{2}
37 1+6.24iT−37T2 1 + 6.24iT - 37T^{2}
41 1+3.07T+41T2 1 + 3.07T + 41T^{2}
43 1−9.41T+43T2 1 - 9.41T + 43T^{2}
47 1−8.82iT−47T2 1 - 8.82iT - 47T^{2}
53 1−8.82T+53T2 1 - 8.82T + 53T^{2}
59 1−7.17T+59T2 1 - 7.17T + 59T^{2}
61 1−12.4T+61T2 1 - 12.4T + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+2.82T+71T2 1 + 2.82T + 71T^{2}
73 1+14.4T+73T2 1 + 14.4T + 73T^{2}
79 1+9.31iT−79T2 1 + 9.31iT - 79T^{2}
83 1+7.17iT−83T2 1 + 7.17iT - 83T^{2}
89 1+13.4T+89T2 1 + 13.4T + 89T^{2}
97 1−9.41iT−97T2 1 - 9.41iT - 97T^{2}
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   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

−10.46806142761083857396304238053, −10.10049205314492014644531484981, −8.823480936306068995985327888705, −8.297655785930812812975445660672, −7.27667189790026424773862825380, −5.98540694012270788031111082900, −5.03887163466081568366153761917, −4.30541185027600792492673443088, −2.58225121220856977310922473630, −0.804001058421859420290469090962, 1.35657485493794321942914727266, 2.33839329309589272615354702397, 4.22122000047918702823790887336, 5.50175612034016594296986682554, 6.52713972028414602226997519436, 7.14946053955124206373481033098, 8.277263154047694248599380089730, 8.658180644461699747566894654653, 10.10605250084482035912886282758, 10.99044168036423299835057418374

Graph of the ZZ-function along the critical line