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Properties

Label 2-9702-1.1-c1-0-14
Degree 22
Conductor 97029702
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.585·5-s − 8-s − 0.585·10-s − 11-s − 3.82·13-s + 16-s − 3.65·17-s − 0.585·19-s + 0.585·20-s + 22-s + 6.24·23-s − 4.65·25-s + 3.82·26-s − 2.65·29-s − 4·31-s − 32-s + 3.65·34-s − 9.41·37-s + 0.585·38-s − 0.585·40-s + 5.41·41-s − 5.65·43-s − 44-s − 6.24·46-s + 10.4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.261·5-s − 0.353·8-s − 0.185·10-s − 0.301·11-s − 1.06·13-s + 0.250·16-s − 0.886·17-s − 0.134·19-s + 0.130·20-s + 0.213·22-s + 1.30·23-s − 0.931·25-s + 0.750·26-s − 0.493·29-s − 0.718·31-s − 0.176·32-s + 0.627·34-s − 1.54·37-s + 0.0950·38-s − 0.0926·40-s + 0.845·41-s − 0.862·43-s − 0.150·44-s − 0.920·46-s + 1.52·47-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: 11
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: 00
Selberg data: (2, 9702, ( :1/2), 1)(2,\ 9702,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.94144015840.9414401584
L(12)L(\frac12) ≈\approx 0.94144015840.9414401584
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 1−0.585T+5T2 1 - 0.585T + 5T^{2}
13 1+3.82T+13T2 1 + 3.82T + 13T^{2}
17 1+3.65T+17T2 1 + 3.65T + 17T^{2}
19 1+0.585T+19T2 1 + 0.585T + 19T^{2}
23 1−6.24T+23T2 1 - 6.24T + 23T^{2}
29 1+2.65T+29T2 1 + 2.65T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+9.41T+37T2 1 + 9.41T + 37T^{2}
41 1−5.41T+41T2 1 - 5.41T + 41T^{2}
43 1+5.65T+43T2 1 + 5.65T + 43T^{2}
47 1−10.4T+47T2 1 - 10.4T + 47T^{2}
53 1+7.89T+53T2 1 + 7.89T + 53T^{2}
59 1−5.58T+59T2 1 - 5.58T + 59T^{2}
61 1−11.8T+61T2 1 - 11.8T + 61T^{2}
67 1−2.75T+67T2 1 - 2.75T + 67T^{2}
71 1−11.0T+71T2 1 - 11.0T + 71T^{2}
73 1+9.41T+73T2 1 + 9.41T + 73T^{2}
79 1+13.2T+79T2 1 + 13.2T + 79T^{2}
83 1−12.1T+83T2 1 - 12.1T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1+3.82T+97T2 1 + 3.82T + 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

−7.51183389471990926566209118894, −7.19576411792399136382220659439, −6.51882453803484115696885308398, −5.60378833607157166717260971034, −5.09245245511981730746161516684, −4.18720382324560450632702719503, −3.25167360235717898908621419292, −2.36802872553764179731784778409, −1.80070410638963361008062689809, −0.49362482980919079971154421053, 0.49362482980919079971154421053, 1.80070410638963361008062689809, 2.36802872553764179731784778409, 3.25167360235717898908621419292, 4.18720382324560450632702719503, 5.09245245511981730746161516684, 5.60378833607157166717260971034, 6.51882453803484115696885308398, 7.19576411792399136382220659439, 7.51183389471990926566209118894

Graph of the ZZ-function along the critical line