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Properties

Label 2-6800-1.1-c1-0-121
Degree 22
Conductor 68006800
Sign −1-1
Analytic cond. 54.298254.2982
Root an. cond. 7.368737.36873
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  + 1.23·3-s − 1.23·7-s − 1.47·9-s + 1.23·11-s − 4.47·13-s − 17-s + 6.47·19-s − 1.52·21-s − 1.23·23-s − 5.52·27-s + 2·29-s − 1.23·31-s + 1.52·33-s + 10.9·37-s − 5.52·39-s + 2·41-s − 1.52·43-s + 12.9·47-s − 5.47·49-s − 1.23·51-s + 2·53-s + 8.00·57-s − 14.4·59-s + 6.94·61-s + 1.81·63-s − 12·67-s − 1.52·69-s + ⋯
L(s)  = 1  + 0.713·3-s − 0.467·7-s − 0.490·9-s + 0.372·11-s − 1.24·13-s − 0.242·17-s + 1.48·19-s − 0.333·21-s − 0.257·23-s − 1.06·27-s + 0.371·29-s − 0.222·31-s + 0.265·33-s + 1.79·37-s − 0.885·39-s + 0.312·41-s − 0.232·43-s + 1.88·47-s − 0.781·49-s − 0.173·51-s + 0.274·53-s + 1.05·57-s − 1.88·59-s + 0.889·61-s + 0.229·63-s − 1.46·67-s − 0.183·69-s + ⋯

Functional equation

Λ(s)=(6800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 68006800    =    24⋅52⋅172^{4} \cdot 5^{2} \cdot 17
Sign: −1-1
Analytic conductor: 54.298254.2982
Root analytic conductor: 7.368737.36873
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6800, ( :1/2), −1)(2,\ 6800,\ (\ :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 1
5 1 1
17 1+T 1 + T
good3 1−1.23T+3T2 1 - 1.23T + 3T^{2}
7 1+1.23T+7T2 1 + 1.23T + 7T^{2}
11 1−1.23T+11T2 1 - 1.23T + 11T^{2}
13 1+4.47T+13T2 1 + 4.47T + 13T^{2}
19 1−6.47T+19T2 1 - 6.47T + 19T^{2}
23 1+1.23T+23T2 1 + 1.23T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+1.23T+31T2 1 + 1.23T + 31T^{2}
37 1−10.9T+37T2 1 - 10.9T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+1.52T+43T2 1 + 1.52T + 43T^{2}
47 1−12.9T+47T2 1 - 12.9T + 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1+14.4T+59T2 1 + 14.4T + 59T^{2}
61 1−6.94T+61T2 1 - 6.94T + 61T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1+9.23T+71T2 1 + 9.23T + 71T^{2}
73 1+14.9T+73T2 1 + 14.9T + 73T^{2}
79 1+11.7T+79T2 1 + 11.7T + 79T^{2}
83 1−1.52T+83T2 1 - 1.52T + 83T^{2}
89 1+7.52T+89T2 1 + 7.52T + 89T^{2}
97 1+2T+97T2 1 + 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

−7.53061266941886125422754389836, −7.21080385332108418266779622024, −6.13372400624271500861122358528, −5.60715061668560898884457265062, −4.65648314700274409299220172172, −3.92397123323433387355684070918, −2.86015258418388898593744428251, −2.68025832639107454132540058812, −1.36968887020875132072044389145, 0, 1.36968887020875132072044389145, 2.68025832639107454132540058812, 2.86015258418388898593744428251, 3.92397123323433387355684070918, 4.65648314700274409299220172172, 5.60715061668560898884457265062, 6.13372400624271500861122358528, 7.21080385332108418266779622024, 7.53061266941886125422754389836

Graph of the ZZ-function along the critical line