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Properties

Label 2-9702-1.1-c1-0-11
Degree 22
Conductor 97029702
Sign 11
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 11-s − 2·13-s + 16-s + 17-s + 3·19-s − 2·20-s + 22-s + 23-s − 25-s + 2·26-s + 29-s + 2·31-s − 32-s − 34-s − 5·37-s − 3·38-s + 2·40-s − 10·41-s + 43-s − 44-s − 46-s + 7·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.688·19-s − 0.447·20-s + 0.213·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.185·29-s + 0.359·31-s − 0.176·32-s − 0.171·34-s − 0.821·37-s − 0.486·38-s + 0.316·40-s − 1.56·41-s + 0.152·43-s − 0.150·44-s − 0.147·46-s + 1.02·47-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: yes
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 0.77573592580.7757359258
L(12)L(\frac12) ≈\approx 0.77573592580.7757359258
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+2T+pT2 1 + 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−T+pT2 1 - T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+7T+pT2 1 + 7 T + p T^{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

−7.85747770802078815545372392454, −7.07460183022210449527023211966, −6.65765666986493420660734961996, −5.56720138691284046709681890077, −5.02902822390966345594923202582, −4.07525073211865380356777462168, −3.34421680996131307135244884602, −2.59569625731930446489042912023, −1.56672594819181568856264595472, −0.47141604315454646824999418909, 0.47141604315454646824999418909, 1.56672594819181568856264595472, 2.59569625731930446489042912023, 3.34421680996131307135244884602, 4.07525073211865380356777462168, 5.02902822390966345594923202582, 5.56720138691284046709681890077, 6.65765666986493420660734961996, 7.07460183022210449527023211966, 7.85747770802078815545372392454

Graph of the ZZ-function along the critical line