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Properties

Label 2-9280-1.1-c1-0-214
Degree 22
Conductor 92809280
Sign −1-1
Analytic cond. 74.101174.1011
Root an. cond. 8.608208.60820
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·3-s + 5-s + 3.85·7-s − 0.381·9-s − 1.23·11-s − 6.09·13-s + 1.61·15-s − 1.38·17-s − 7.23·19-s + 6.23·21-s − 0.854·23-s + 25-s − 5.47·27-s − 29-s + 0.618·31-s − 2.00·33-s + 3.85·35-s − 4.76·37-s − 9.85·39-s + 9.70·41-s + 5.38·43-s − 0.381·45-s − 8·47-s + 7.85·49-s − 2.23·51-s + 6.32·53-s − 1.23·55-s + ⋯
L(s)  = 1  + 0.934·3-s + 0.447·5-s + 1.45·7-s − 0.127·9-s − 0.372·11-s − 1.68·13-s + 0.417·15-s − 0.335·17-s − 1.66·19-s + 1.36·21-s − 0.178·23-s + 0.200·25-s − 1.05·27-s − 0.185·29-s + 0.111·31-s − 0.348·33-s + 0.651·35-s − 0.783·37-s − 1.57·39-s + 1.51·41-s + 0.820·43-s − 0.0569·45-s − 1.16·47-s + 1.12·49-s − 0.313·51-s + 0.868·53-s − 0.166·55-s + ⋯

Functional equation

Λ(s)=(9280s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9280s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92809280    =    26⋅5⋅292^{6} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 74.101174.1011
Root analytic conductor: 8.608208.60820
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9280, ( :1/2), −1)(2,\ 9280,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
5 1−T 1 - T
29 1+T 1 + T
good3 1−1.61T+3T2 1 - 1.61T + 3T^{2}
7 1−3.85T+7T2 1 - 3.85T + 7T^{2}
11 1+1.23T+11T2 1 + 1.23T + 11T^{2}
13 1+6.09T+13T2 1 + 6.09T + 13T^{2}
17 1+1.38T+17T2 1 + 1.38T + 17T^{2}
19 1+7.23T+19T2 1 + 7.23T + 19T^{2}
23 1+0.854T+23T2 1 + 0.854T + 23T^{2}
31 1−0.618T+31T2 1 - 0.618T + 31T^{2}
37 1+4.76T+37T2 1 + 4.76T + 37T^{2}
41 1−9.70T+41T2 1 - 9.70T + 41T^{2}
43 1−5.38T+43T2 1 - 5.38T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1−6.32T+53T2 1 - 6.32T + 53T^{2}
59 1−11.6T+59T2 1 - 11.6T + 59T^{2}
61 1+8.85T+61T2 1 + 8.85T + 61T^{2}
67 1+6.47T+67T2 1 + 6.47T + 67T^{2}
71 1+4.94T+71T2 1 + 4.94T + 71T^{2}
73 1−13.0T+73T2 1 - 13.0T + 73T^{2}
79 1+14.0T+79T2 1 + 14.0T + 79T^{2}
83 1+2.29T+83T2 1 + 2.29T + 83T^{2}
89 1−11.7T+89T2 1 - 11.7T + 89T^{2}
97 1+15.7T+97T2 1 + 15.7T + 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

−7.55206064723423335593254502470, −6.89214496612914423220694980347, −5.89588607269668331921111960380, −5.20932837965744016296418115028, −4.57100548365004783540512273773, −3.94263195039716453567812476071, −2.58686133776375308009105927657, −2.40332193047745199439211773288, −1.59483722820858025670703466595, 0, 1.59483722820858025670703466595, 2.40332193047745199439211773288, 2.58686133776375308009105927657, 3.94263195039716453567812476071, 4.57100548365004783540512273773, 5.20932837965744016296418115028, 5.89588607269668331921111960380, 6.89214496612914423220694980347, 7.55206064723423335593254502470

Graph of the ZZ-function along the critical line