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Properties

Label 2-9702-1.1-c1-0-135
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 + 2.74·5-s − 8-s − 2.74·10-s − 11-s − 5.49·13-s + 16-s + 17-s + 8.03·19-s + 2.74·20-s + 22-s + 1.29·23-s + 2.54·25-s + 5.49·26-s − 4.54·29-s − 5.08·31-s − 32-s − 34-s − 4.03·37-s − 8.03·38-s − 2.74·40-s + 5.54·41-s − 4.03·43-s − 44-s − 1.29·46-s − 9.29·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.22·5-s − 0.353·8-s − 0.868·10-s − 0.301·11-s − 1.52·13-s + 0.250·16-s + 0.242·17-s + 1.84·19-s + 0.614·20-s + 0.213·22-s + 0.269·23-s + 0.508·25-s + 1.07·26-s − 0.843·29-s − 0.913·31-s − 0.176·32-s − 0.171·34-s − 0.663·37-s − 1.30·38-s − 0.434·40-s + 0.865·41-s − 0.615·43-s − 0.150·44-s − 0.190·46-s − 1.35·47-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: −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 1−2.74T+5T2 1 - 2.74T + 5T^{2}
13 1+5.49T+13T2 1 + 5.49T + 13T^{2}
17 1−T+17T2 1 - T + 17T^{2}
19 1−8.03T+19T2 1 - 8.03T + 19T^{2}
23 1−1.29T+23T2 1 - 1.29T + 23T^{2}
29 1+4.54T+29T2 1 + 4.54T + 29T^{2}
31 1+5.08T+31T2 1 + 5.08T + 31T^{2}
37 1+4.03T+37T2 1 + 4.03T + 37T^{2}
41 1−5.54T+41T2 1 - 5.54T + 41T^{2}
43 1+4.03T+43T2 1 + 4.03T + 43T^{2}
47 1+9.29T+47T2 1 + 9.29T + 47T^{2}
53 1+5.49T+53T2 1 + 5.49T + 53T^{2}
59 1+9.52T+59T2 1 + 9.52T + 59T^{2}
61 1−1.65T+61T2 1 - 1.65T + 61T^{2}
67 1−3.54T+67T2 1 - 3.54T + 67T^{2}
71 1+2.54T+71T2 1 + 2.54T + 71T^{2}
73 1−8.58T+73T2 1 - 8.58T + 73T^{2}
79 1−12.2T+79T2 1 - 12.2T + 79T^{2}
83 1+14.5T+83T2 1 + 14.5T + 83T^{2}
89 1+6.50T+89T2 1 + 6.50T + 89T^{2}
97 1−0.0872T+97T2 1 - 0.0872T + 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

−7.52549781412875013337099053754, −6.77585713292461089085449624996, −5.99478621779306996641276815298, −5.25307533733522518018502556836, −4.97576629217043130456938403403, −3.54326963820004365151583007326, −2.80186771278151858170541182136, −2.04472049630453939729398549057, −1.31480939060367634484582915095, 0, 1.31480939060367634484582915095, 2.04472049630453939729398549057, 2.80186771278151858170541182136, 3.54326963820004365151583007326, 4.97576629217043130456938403403, 5.25307533733522518018502556836, 5.99478621779306996641276815298, 6.77585713292461089085449624996, 7.52549781412875013337099053754

Graph of the ZZ-function along the critical line