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Properties

Label 2-1472-1.1-c3-0-117
Degree 22
Conductor 14721472
Sign −1-1
Analytic cond. 86.850886.8508
Root an. cond. 9.319379.31937
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 6·5-s − 8·7-s − 2·9-s − 34·11-s + 57·13-s + 30·15-s − 80·17-s + 70·19-s − 40·21-s + 23·23-s − 89·25-s − 145·27-s − 245·29-s + 103·31-s − 170·33-s − 48·35-s + 298·37-s + 285·39-s + 95·41-s − 88·43-s − 12·45-s − 357·47-s − 279·49-s − 400·51-s + 414·53-s − 204·55-s + ⋯
L(s)  = 1  + 0.962·3-s + 0.536·5-s − 0.431·7-s − 0.0740·9-s − 0.931·11-s + 1.21·13-s + 0.516·15-s − 1.14·17-s + 0.845·19-s − 0.415·21-s + 0.208·23-s − 0.711·25-s − 1.03·27-s − 1.56·29-s + 0.596·31-s − 0.896·33-s − 0.231·35-s + 1.32·37-s + 1.17·39-s + 0.361·41-s − 0.312·43-s − 0.0397·45-s − 1.10·47-s − 0.813·49-s − 1.09·51-s + 1.07·53-s − 0.500·55-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: −1-1
Analytic conductor: 86.850886.8508
Root analytic conductor: 9.319379.31937
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1472, ( :3/2), −1)(2,\ 1472,\ (\ :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(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
23 1−pT 1 - p T
good3 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
5 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
7 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
13 1−57T+p3T2 1 - 57 T + p^{3} T^{2}
17 1+80T+p3T2 1 + 80 T + p^{3} T^{2}
19 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
29 1+245T+p3T2 1 + 245 T + p^{3} T^{2}
31 1−103T+p3T2 1 - 103 T + p^{3} T^{2}
37 1−298T+p3T2 1 - 298 T + p^{3} T^{2}
41 1−95T+p3T2 1 - 95 T + p^{3} T^{2}
43 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
47 1+357T+p3T2 1 + 357 T + p^{3} T^{2}
53 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
59 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
61 1+822T+p3T2 1 + 822 T + p^{3} T^{2}
67 1+926T+p3T2 1 + 926 T + p^{3} T^{2}
71 1−335T+p3T2 1 - 335 T + p^{3} T^{2}
73 1+899T+p3T2 1 + 899 T + p^{3} T^{2}
79 1+1322T+p3T2 1 + 1322 T + p^{3} T^{2}
83 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
89 1+460T+p3T2 1 + 460 T + p^{3} T^{2}
97 1+964T+p3T2 1 + 964 T + p^{3} 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

−8.788876249294354183927710988854, −8.047375468163031405367381312777, −7.29474836369551728543653654538, −6.14752426239874499050710473682, −5.61023529658559219712714891186, −4.33903202789494036501720708804, −3.32434363589712739954045562743, −2.60927413533643322818380708542, −1.60568297861802847785950716483, 0, 1.60568297861802847785950716483, 2.60927413533643322818380708542, 3.32434363589712739954045562743, 4.33903202789494036501720708804, 5.61023529658559219712714891186, 6.14752426239874499050710473682, 7.29474836369551728543653654538, 8.047375468163031405367381312777, 8.788876249294354183927710988854

Graph of the ZZ-function along the critical line