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Properties

Label 2-10608-1.1-c1-0-10
Degree 22
Conductor 1060810608
Sign 11
Analytic cond. 84.705384.7053
Root an. cond. 9.203549.20354
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 + 2·5-s + 4·7-s + 9-s − 2·11-s − 13-s + 2·15-s + 17-s − 2·19-s + 4·21-s + 8·23-s − 25-s + 27-s + 2·29-s − 4·31-s − 2·33-s + 8·35-s + 8·37-s − 39-s − 10·41-s + 2·45-s + 8·47-s + 9·49-s + 51-s − 2·53-s − 4·55-s − 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.458·19-s + 0.872·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.348·33-s + 1.35·35-s + 1.31·37-s − 0.160·39-s − 1.56·41-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.140·51-s − 0.274·53-s − 0.539·55-s − 0.264·57-s + ⋯

Functional equation

Λ(s)=(10608s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10608s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1060810608    =    24⋅3⋅13⋅172^{4} \cdot 3 \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 84.705384.7053
Root analytic conductor: 9.203549.20354
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 10608, ( :1/2), 1)(2,\ 10608,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0705795554.070579555
L(12)L(\frac12) ≈\approx 4.0705795554.070579555
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
13 1+T 1 + T
17 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+16T+pT2 1 + 16 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+4T+pT2 1 + 4 T + p T^{2}
97 1−12T+pT2 1 - 12 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

−16.69399460950926, −15.92510910343638, −15.08142001758183, −14.87696132168192, −14.33004156074182, −13.71162358151637, −13.22442744166091, −12.72179806461006, −11.87135367894897, −11.28122395654848, −10.63471584288408, −10.19694615582503, −9.395097915084181, −8.873593748935786, −8.249571057472508, −7.675566955683405, −7.098645096824280, −6.235879324759636, −5.361940539231294, −4.993923917838953, −4.271589756166169, −3.247302397357388, −2.380312787102766, −1.870308439624586, −0.9590617211748903, 0.9590617211748903, 1.870308439624586, 2.380312787102766, 3.247302397357388, 4.271589756166169, 4.993923917838953, 5.361940539231294, 6.235879324759636, 7.098645096824280, 7.675566955683405, 8.249571057472508, 8.873593748935786, 9.395097915084181, 10.19694615582503, 10.63471584288408, 11.28122395654848, 11.87135367894897, 12.72179806461006, 13.22442744166091, 13.71162358151637, 14.33004156074182, 14.87696132168192, 15.08142001758183, 15.92510910343638, 16.69399460950926

Graph of the ZZ-function along the critical line