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Properties

Label 2-768-1.1-c1-0-6
Degree 22
Conductor 768768
Sign 11
Analytic cond. 6.132516.13251
Root an. cond. 2.476392.47639
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  + 3-s + 4·7-s + 9-s − 4·11-s + 4·13-s − 2·17-s + 4·19-s + 4·21-s + 8·23-s − 5·25-s + 27-s − 8·29-s + 4·31-s − 4·33-s − 4·37-s + 4·39-s + 6·41-s − 4·43-s + 8·47-s + 9·49-s − 2·51-s − 8·53-s + 4·57-s + 12·59-s + 12·61-s + 4·63-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.872·21-s + 1.66·23-s − 25-s + 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.696·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.09·53-s + 0.529·57-s + 1.56·59-s + 1.53·61-s + 0.503·63-s − 1.46·67-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: 11
Analytic conductor: 6.132516.13251
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 768, ( :1/2), 1)(2,\ 768,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1726561662.172656166
L(12)L(\frac12) ≈\approx 2.1726561662.172656166
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
good5 1+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+8T+pT2 1 + 8 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−10.46186321886751362762916978052, −9.298383051990469888672283644214, −8.520382921239941720035881428633, −7.84363031960006724677694678337, −7.15944714304425705232693435926, −5.67206581946498649675335111392, −4.95203593728251161575038294540, −3.84720863319685552465873529850, −2.59900951660796699369904554010, −1.39752451983425368394719286578, 1.39752451983425368394719286578, 2.59900951660796699369904554010, 3.84720863319685552465873529850, 4.95203593728251161575038294540, 5.67206581946498649675335111392, 7.15944714304425705232693435926, 7.84363031960006724677694678337, 8.520382921239941720035881428633, 9.298383051990469888672283644214, 10.46186321886751362762916978052

Graph of the ZZ-function along the critical line