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Properties

Label 2-9702-1.1-c1-0-142
Degree 22
Conductor 97029702
Sign −1-1
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
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 − 11-s + 5.65·13-s + 16-s + 1.41·17-s − 4.24·19-s − 1.41·20-s − 22-s − 4·23-s − 2.99·25-s + 5.65·26-s − 4.24·31-s + 32-s + 1.41·34-s − 6·37-s − 4.24·38-s − 1.41·40-s + 4.24·41-s − 4·43-s − 44-s − 4·46-s − 1.41·47-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 − 0.301·11-s + 1.56·13-s + 0.250·16-s + 0.342·17-s − 0.973·19-s − 0.316·20-s − 0.213·22-s − 0.834·23-s − 0.599·25-s + 1.10·26-s − 0.762·31-s + 0.176·32-s + 0.242·34-s − 0.986·37-s − 0.688·38-s − 0.223·40-s + 0.662·41-s − 0.609·43-s − 0.150·44-s − 0.589·46-s − 0.206·47-s − 0.424·50-s + ⋯

Functional equation

Λ(s)=(9702s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9702s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 97029702    =    2⋅32⋅72⋅112 \cdot 3^{2} \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 77.470877.4708
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9702, ( :1/2), −1)(2,\ 9702,\ (\ :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
11 1+T 1 + T
good5 1+1.41T+5T2 1 + 1.41T + 5T^{2}
13 1−5.65T+13T2 1 - 5.65T + 13T^{2}
17 1−1.41T+17T2 1 - 1.41T + 17T^{2}
19 1+4.24T+19T2 1 + 4.24T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+4.24T+31T2 1 + 4.24T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1−4.24T+41T2 1 - 4.24T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+1.41T+47T2 1 + 1.41T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+2.82T+59T2 1 + 2.82T + 59T^{2}
61 1+5.65T+61T2 1 + 5.65T + 61T^{2}
67 1+10T+67T2 1 + 10T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+4.24T+73T2 1 + 4.24T + 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1−15.5T+83T2 1 - 15.5T + 83T^{2}
89 1−14.1T+89T2 1 - 14.1T + 89T^{2}
97 1+5.65T+97T2 1 + 5.65T + 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

−7.37307542589453890728648680099, −6.43891793780202291204829760769, −6.01691563405721255381535357802, −5.30046086165808430594803602271, −4.40060901485890855012077169607, −3.80414807200140265702729408107, −3.33618775732448230238689043757, −2.23883143576820183124192350105, −1.39114037757314821566517643112, 0, 1.39114037757314821566517643112, 2.23883143576820183124192350105, 3.33618775732448230238689043757, 3.80414807200140265702729408107, 4.40060901485890855012077169607, 5.30046086165808430594803602271, 6.01691563405721255381535357802, 6.43891793780202291204829760769, 7.37307542589453890728648680099

Graph of the ZZ-function along the critical line