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Properties

Label 2-6-1.1-c17-0-2
Degree 22
Conductor 66
Sign −1-1
Analytic cond. 10.993310.9933
Root an. cond. 3.315613.31561
Motivic weight 1717
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 256·2-s − 6.56e3·3-s + 6.55e4·4-s + 6.45e5·5-s + 1.67e6·6-s + 3.97e6·7-s − 1.67e7·8-s + 4.30e7·9-s − 1.65e8·10-s − 5.00e8·11-s − 4.29e8·12-s − 5.42e9·13-s − 1.01e9·14-s − 4.23e9·15-s + 4.29e9·16-s − 5.46e9·17-s − 1.10e10·18-s − 5.38e10·19-s + 4.22e10·20-s − 2.60e10·21-s + 1.28e11·22-s + 5.78e11·23-s + 1.10e11·24-s − 3.46e11·25-s + 1.38e12·26-s − 2.82e11·27-s + 2.60e11·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.738·5-s + 0.408·6-s + 0.260·7-s − 0.353·8-s + 1/3·9-s − 0.522·10-s − 0.703·11-s − 0.288·12-s − 1.84·13-s − 0.184·14-s − 0.426·15-s + 1/4·16-s − 0.190·17-s − 0.235·18-s − 0.727·19-s + 0.369·20-s − 0.150·21-s + 0.497·22-s + 1.54·23-s + 0.204·24-s − 0.454·25-s + 1.30·26-s − 0.192·27-s + 0.130·28-s + ⋯

Functional equation

Λ(s)=(6s/2 ΓC(s) L(s)=(− Λ(18−s)\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}
Λ(s)=(6s/2 ΓC(s+17/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 66    =    2⋅32 \cdot 3
Sign: −1-1
Analytic conductor: 10.993310.9933
Root analytic conductor: 3.315613.31561
Motivic weight: 1717
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6, ( :17/2), −1)(2,\ 6,\ (\ :17/2),\ -1)

Particular Values

L(9)L(9) == 00
L(12)L(\frac12) == 00
L(192)L(\frac{19}{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+p8T 1 + p^{8} T
3 1+p8T 1 + p^{8} T
good5 1−25806p2T+p17T2 1 - 25806 p^{2} T + p^{17} T^{2}
7 1−567776pT+p17T2 1 - 567776 p T + p^{17} T^{2}
11 1+45460788pT+p17T2 1 + 45460788 p T + p^{17} T^{2}
13 1+5425661314T+p17T2 1 + 5425661314 T + p^{17} T^{2}
17 1+5466992958T+p17T2 1 + 5466992958 T + p^{17} T^{2}
19 1+53889877060T+p17T2 1 + 53889877060 T + p^{17} T^{2}
23 1−578906836536T+p17T2 1 - 578906836536 T + p^{17} T^{2}
29 1+4619583681690T+p17T2 1 + 4619583681690 T + p^{17} T^{2}
31 1+6802815567448T+p17T2 1 + 6802815567448 T + p^{17} T^{2}
37 1+19571909422138T+p17T2 1 + 19571909422138 T + p^{17} T^{2}
41 1−57213620756922T+p17T2 1 - 57213620756922 T + p^{17} T^{2}
43 1+24501250225084T+p17T2 1 + 24501250225084 T + p^{17} T^{2}
47 1−184283998832832T+p17T2 1 - 184283998832832 T + p^{17} T^{2}
53 1+206542562280354T+p17T2 1 + 206542562280354 T + p^{17} T^{2}
59 1+418648048246140T+p17T2 1 + 418648048246140 T + p^{17} T^{2}
61 1−2501287878088382T+p17T2 1 - 2501287878088382 T + p^{17} T^{2}
67 1+145692866050948T+p17T2 1 + 145692866050948 T + p^{17} T^{2}
71 1+5364313152664248T+p17T2 1 + 5364313152664248 T + p^{17} T^{2}
73 1−3302058927938186T+p17T2 1 - 3302058927938186 T + p^{17} T^{2}
79 1−22067463278260760T+p17T2 1 - 22067463278260760 T + p^{17} T^{2}
83 1−20438378406354876T+p17T2 1 - 20438378406354876 T + p^{17} T^{2}
89 1+56063805950152710T+p17T2 1 + 56063805950152710 T + p^{17} T^{2}
97 1+118254406396110718T+p17T2 1 + 118254406396110718 T + p^{17} T^{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

−17.68693402654015723782842418374, −16.81530778009832749235234374319, −14.92857252611761500445607255585, −12.75190728140454758704120009201, −10.90766785822369459716757028814, −9.483467487472704116543893836061, −7.28870293225689242567611511459, −5.32472584341642081578733119505, −2.11022976097817694853914365166, 0, 2.11022976097817694853914365166, 5.32472584341642081578733119505, 7.28870293225689242567611511459, 9.483467487472704116543893836061, 10.90766785822369459716757028814, 12.75190728140454758704120009201, 14.92857252611761500445607255585, 16.81530778009832749235234374319, 17.68693402654015723782842418374

Graph of the ZZ-function along the critical line