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Properties

Label 2-912-1.1-c1-0-11
Degree 22
Conductor 912912
Sign −1-1
Analytic cond. 7.282357.28235
Root an. cond. 2.698582.69858
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s + 6·13-s + 2·15-s − 6·17-s + 19-s − 4·23-s − 25-s − 27-s + 2·29-s − 8·31-s − 10·37-s − 6·39-s − 2·41-s + 4·43-s − 2·45-s − 12·47-s − 7·49-s + 6·51-s − 6·53-s − 57-s + 12·59-s − 2·61-s − 12·65-s + 4·67-s + 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.66·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 1.64·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.75·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.132·57-s + 1.56·59-s − 0.256·61-s − 1.48·65-s + 0.488·67-s + 0.481·69-s + ⋯

Functional equation

Λ(s)=(912s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(912s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 912912    =    24⋅3⋅192^{4} \cdot 3 \cdot 19
Sign: −1-1
Analytic conductor: 7.282357.28235
Root analytic conductor: 2.698582.69858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 912, ( :1/2), −1)(2,\ 912,\ (\ :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
3 1+T 1 + T
19 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−9.717867966311693207189859225084, −8.673289841767403383115719575464, −8.099644677533565214444467559710, −6.98236656892345640095319736227, −6.30774150521524867405980638808, −5.28029214515602867786662657712, −4.16838172205224943400021226417, −3.49609663069962101813276746331, −1.72203721066548968493435337820, 0, 1.72203721066548968493435337820, 3.49609663069962101813276746331, 4.16838172205224943400021226417, 5.28029214515602867786662657712, 6.30774150521524867405980638808, 6.98236656892345640095319736227, 8.099644677533565214444467559710, 8.673289841767403383115719575464, 9.717867966311693207189859225084

Graph of the ZZ-function along the critical line