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Properties

Label 2-9702-1.1-c1-0-10
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 − 3.79·5-s − 8-s + 3.79·10-s − 11-s + 0.361·13-s + 16-s + 4.11·17-s + 4.15·19-s − 3.79·20-s + 22-s − 0.542·23-s + 9.42·25-s − 0.361·26-s − 0.767·29-s − 8.80·31-s − 32-s − 4.11·34-s + 2.28·37-s − 4.15·38-s + 3.79·40-s − 9.13·41-s − 10.1·43-s − 44-s + 0.542·46-s + 2.98·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.69·5-s − 0.353·8-s + 1.20·10-s − 0.301·11-s + 0.100·13-s + 0.250·16-s + 0.998·17-s + 0.954·19-s − 0.849·20-s + 0.213·22-s − 0.113·23-s + 1.88·25-s − 0.0707·26-s − 0.142·29-s − 1.58·31-s − 0.176·32-s − 0.705·34-s + 0.375·37-s − 0.674·38-s + 0.600·40-s − 1.42·41-s − 1.55·43-s − 0.150·44-s + 0.0800·46-s + 0.435·47-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 0.70369711290.7036971129
L(12)L(\frac12) ≈\approx 0.70369711290.7036971129
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+3.79T+5T2 1 + 3.79T + 5T^{2}
13 1−0.361T+13T2 1 - 0.361T + 13T^{2}
17 1−4.11T+17T2 1 - 4.11T + 17T^{2}
19 1−4.15T+19T2 1 - 4.15T + 19T^{2}
23 1+0.542T+23T2 1 + 0.542T + 23T^{2}
29 1+0.767T+29T2 1 + 0.767T + 29T^{2}
31 1+8.80T+31T2 1 + 8.80T + 31T^{2}
37 1−2.28T+37T2 1 - 2.28T + 37T^{2}
41 1+9.13T+41T2 1 + 9.13T + 41T^{2}
43 1+10.1T+43T2 1 + 10.1T + 43T^{2}
47 1−2.98T+47T2 1 - 2.98T + 47T^{2}
53 1−12.4T+53T2 1 - 12.4T + 53T^{2}
59 1+2.25T+59T2 1 + 2.25T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1−0.603T+67T2 1 - 0.603T + 67T^{2}
71 1−6.87T+71T2 1 - 6.87T + 71T^{2}
73 1−9.45T+73T2 1 - 9.45T + 73T^{2}
79 1−0.0321T+79T2 1 - 0.0321T + 79T^{2}
83 1+10.8T+83T2 1 + 10.8T + 83T^{2}
89 1−1.29T+89T2 1 - 1.29T + 89T^{2}
97 1+18.2T+97T2 1 + 18.2T + 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

−7.64639561792872286237024838392, −7.27778828988915799128135572717, −6.66564050193946044624245464395, −5.50361048404764323280807173334, −5.06207505048478503417029193598, −3.82658408903865063364265524386, −3.57831477886499631822968295562, −2.67332531852282836318390317425, −1.44543148355820861875495028477, −0.46830935610164087318279491479, 0.46830935610164087318279491479, 1.44543148355820861875495028477, 2.67332531852282836318390317425, 3.57831477886499631822968295562, 3.82658408903865063364265524386, 5.06207505048478503417029193598, 5.50361048404764323280807173334, 6.66564050193946044624245464395, 7.27778828988915799128135572717, 7.64639561792872286237024838392

Graph of the ZZ-function along the critical line