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Properties

Label 2-882-1.1-c1-0-13
Degree 22
Conductor 882882
Sign −1-1
Analytic cond. 7.042807.04280
Root an. cond. 2.653822.65382
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  − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 4·11-s + 4.24·13-s + 16-s + 7.07·17-s − 5.65·19-s − 1.41·20-s + 4·22-s − 8·23-s − 2.99·25-s − 4.24·26-s − 2·29-s − 32-s − 7.07·34-s + 4·37-s + 5.65·38-s + 1.41·40-s − 9.89·41-s − 4·43-s − 4·44-s + 8·46-s − 5.65·47-s + 2.99·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.632·5-s − 0.353·8-s + 0.447·10-s − 1.20·11-s + 1.17·13-s + 0.250·16-s + 1.71·17-s − 1.29·19-s − 0.316·20-s + 0.852·22-s − 1.66·23-s − 0.599·25-s − 0.832·26-s − 0.371·29-s − 0.176·32-s − 1.21·34-s + 0.657·37-s + 0.917·38-s + 0.223·40-s − 1.54·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s − 0.825·47-s + 0.424·50-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 882, ( :1/2), −1)(2,\ 882,\ (\ :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+T 1 + T
3 1 1
7 1 1
good5 1+1.41T+5T2 1 + 1.41T + 5T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1−4.24T+13T2 1 - 4.24T + 13T^{2}
17 1−7.07T+17T2 1 - 7.07T + 17T^{2}
19 1+5.65T+19T2 1 + 5.65T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1+9.89T+41T2 1 + 9.89T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+5.65T+47T2 1 + 5.65T + 47T^{2}
53 1+4T+53T2 1 + 4T + 53T^{2}
59 1−11.3T+59T2 1 - 11.3T + 59T^{2}
61 1−1.41T+61T2 1 - 1.41T + 61T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+15.5T+73T2 1 + 15.5T + 73T^{2}
79 1+16T+79T2 1 + 16T + 79T^{2}
83 1−5.65T+83T2 1 - 5.65T + 83T^{2}
89 1+7.07T+89T2 1 + 7.07T + 89T^{2}
97 1−7.07T+97T2 1 - 7.07T + 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

−10.01720571963695241720881608827, −8.589941487088499416010561886341, −8.118049983618376092738830727856, −7.50287723561590050483594281799, −6.27011586922826886014853818576, −5.52792116357709787421509266144, −4.11473716940554774137568040594, −3.15874985183319832516167178509, −1.73758245242588432787724477923, 0, 1.73758245242588432787724477923, 3.15874985183319832516167178509, 4.11473716940554774137568040594, 5.52792116357709787421509266144, 6.27011586922826886014853818576, 7.50287723561590050483594281799, 8.118049983618376092738830727856, 8.589941487088499416010561886341, 10.01720571963695241720881608827

Graph of the ZZ-function along the critical line