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Properties

Label 2-9702-1.1-c1-0-113
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 − 1.23·5-s − 8-s + 1.23·10-s − 11-s + 3.23·13-s + 16-s + 2.47·17-s + 7.23·19-s − 1.23·20-s + 22-s − 4·23-s − 3.47·25-s − 3.23·26-s − 4.47·29-s − 2·31-s − 32-s − 2.47·34-s + 6.94·37-s − 7.23·38-s + 1.23·40-s − 2.47·41-s − 10.4·43-s − 44-s + 4·46-s − 2·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.552·5-s − 0.353·8-s + 0.390·10-s − 0.301·11-s + 0.897·13-s + 0.250·16-s + 0.599·17-s + 1.66·19-s − 0.276·20-s + 0.213·22-s − 0.834·23-s − 0.694·25-s − 0.634·26-s − 0.830·29-s − 0.359·31-s − 0.176·32-s − 0.423·34-s + 1.14·37-s − 1.17·38-s + 0.195·40-s − 0.386·41-s − 1.59·43-s − 0.150·44-s + 0.589·46-s − 0.291·47-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+1.23T+5T2 1 + 1.23T + 5T^{2}
13 1−3.23T+13T2 1 - 3.23T + 13T^{2}
17 1−2.47T+17T2 1 - 2.47T + 17T^{2}
19 1−7.23T+19T2 1 - 7.23T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+4.47T+29T2 1 + 4.47T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1−6.94T+37T2 1 - 6.94T + 37T^{2}
41 1+2.47T+41T2 1 + 2.47T + 41T^{2}
43 1+10.4T+43T2 1 + 10.4T + 43T^{2}
47 1+2T+47T2 1 + 2T + 47T^{2}
53 1+8.47T+53T2 1 + 8.47T + 53T^{2}
59 1−2.76T+59T2 1 - 2.76T + 59T^{2}
61 1−0.763T+61T2 1 - 0.763T + 61T^{2}
67 1−11.4T+67T2 1 - 11.4T + 67T^{2}
71 1+6.47T+71T2 1 + 6.47T + 71T^{2}
73 1+12.9T+73T2 1 + 12.9T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+12.1T+83T2 1 + 12.1T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+12.4T+97T2 1 + 12.4T + 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

−7.50981064104359983985970421638, −6.87510090786233085536772419350, −5.92388809734210315233999904073, −5.52652317050472845038906012765, −4.50968305964953760476989565570, −3.55777187437865458875377950042, −3.16579956403131912652733825513, −1.95391792339217301259235669777, −1.13443543830627442846102815885, 0, 1.13443543830627442846102815885, 1.95391792339217301259235669777, 3.16579956403131912652733825513, 3.55777187437865458875377950042, 4.50968305964953760476989565570, 5.52652317050472845038906012765, 5.92388809734210315233999904073, 6.87510090786233085536772419350, 7.50981064104359983985970421638

Graph of the ZZ-function along the critical line