Properties

Label 2-968-1.1-c3-0-28
Degree 22
Conductor 968968
Sign 1-1
Analytic cond. 57.113857.1138
Root an. cond. 7.557377.55737
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.64·3-s − 12.1·5-s − 33.8·7-s + 4.91·9-s + 20.6·13-s + 68.7·15-s + 60.1·17-s + 105.·19-s + 191.·21-s + 186.·23-s + 23.2·25-s + 124.·27-s − 135.·29-s + 89.6·31-s + 412.·35-s − 183.·37-s − 116.·39-s − 372.·41-s − 294.·43-s − 59.8·45-s + 8.47·47-s + 802.·49-s − 339.·51-s − 219.·53-s − 598.·57-s + 260.·59-s + 514.·61-s + ⋯
L(s)  = 1  − 1.08·3-s − 1.08·5-s − 1.82·7-s + 0.181·9-s + 0.440·13-s + 1.18·15-s + 0.858·17-s + 1.27·19-s + 1.98·21-s + 1.69·23-s + 0.185·25-s + 0.889·27-s − 0.867·29-s + 0.519·31-s + 1.98·35-s − 0.817·37-s − 0.479·39-s − 1.41·41-s − 1.04·43-s − 0.198·45-s + 0.0263·47-s + 2.33·49-s − 0.933·51-s − 0.569·53-s − 1.39·57-s + 0.574·59-s + 1.08·61-s + ⋯

Functional equation

Λ(s)=(968s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(968s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    231122^{3} \cdot 11^{2}
Sign: 1-1
Analytic conductor: 57.113857.1138
Root analytic conductor: 7.557377.55737
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 968, ( :3/2), 1)(2,\ 968,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
11 1 1
good3 1+5.64T+27T2 1 + 5.64T + 27T^{2}
5 1+12.1T+125T2 1 + 12.1T + 125T^{2}
7 1+33.8T+343T2 1 + 33.8T + 343T^{2}
13 120.6T+2.19e3T2 1 - 20.6T + 2.19e3T^{2}
17 160.1T+4.91e3T2 1 - 60.1T + 4.91e3T^{2}
19 1105.T+6.85e3T2 1 - 105.T + 6.85e3T^{2}
23 1186.T+1.21e4T2 1 - 186.T + 1.21e4T^{2}
29 1+135.T+2.43e4T2 1 + 135.T + 2.43e4T^{2}
31 189.6T+2.97e4T2 1 - 89.6T + 2.97e4T^{2}
37 1+183.T+5.06e4T2 1 + 183.T + 5.06e4T^{2}
41 1+372.T+6.89e4T2 1 + 372.T + 6.89e4T^{2}
43 1+294.T+7.95e4T2 1 + 294.T + 7.95e4T^{2}
47 18.47T+1.03e5T2 1 - 8.47T + 1.03e5T^{2}
53 1+219.T+1.48e5T2 1 + 219.T + 1.48e5T^{2}
59 1260.T+2.05e5T2 1 - 260.T + 2.05e5T^{2}
61 1514.T+2.26e5T2 1 - 514.T + 2.26e5T^{2}
67 1+263.T+3.00e5T2 1 + 263.T + 3.00e5T^{2}
71 1+309.T+3.57e5T2 1 + 309.T + 3.57e5T^{2}
73 1210.T+3.89e5T2 1 - 210.T + 3.89e5T^{2}
79 1934.T+4.93e5T2 1 - 934.T + 4.93e5T^{2}
83 1238.T+5.71e5T2 1 - 238.T + 5.71e5T^{2}
89 1+620.T+7.04e5T2 1 + 620.T + 7.04e5T^{2}
97 11.20e3T+9.12e5T2 1 - 1.20e3T + 9.12e5T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.367698632416796455037964875748, −8.374630339502662723126232512963, −7.22098374619536424239544202892, −6.71203552661881784892388461265, −5.75257980302797775298357908967, −5.00273773875573917296600761384, −3.53735200960973284239642405761, −3.18748682264423219084776720327, −0.898603883452229434451006403324, 0, 0.898603883452229434451006403324, 3.18748682264423219084776720327, 3.53735200960973284239642405761, 5.00273773875573917296600761384, 5.75257980302797775298357908967, 6.71203552661881784892388461265, 7.22098374619536424239544202892, 8.374630339502662723126232512963, 9.367698632416796455037964875748

Graph of the ZZ-function along the critical line