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Properties

Label 2-936-1.1-c1-0-1
Degree 22
Conductor 936936
Sign 11
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
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  − 4·5-s + 2·11-s − 13-s − 2·17-s + 8·19-s − 4·23-s + 11·25-s + 6·29-s − 4·31-s + 6·37-s + 12·41-s + 4·43-s + 6·47-s − 7·49-s + 2·53-s − 8·55-s + 14·59-s + 10·61-s + 4·65-s − 4·67-s − 2·71-s − 2·73-s − 8·79-s − 14·83-s + 8·85-s − 32·95-s − 10·97-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.603·11-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 11/5·25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s + 1.87·41-s + 0.609·43-s + 0.875·47-s − 49-s + 0.274·53-s − 1.07·55-s + 1.82·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.237·71-s − 0.234·73-s − 0.900·79-s − 1.53·83-s + 0.867·85-s − 3.28·95-s − 1.01·97-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 1)(2,\ 936,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0958963841.095896384
L(12)L(\frac12) ≈\approx 1.0958963841.095896384
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 1
13 1+T 1 + T
good5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+14T+pT2 1 + 14 T + p T^{2}
89 1+pT2 1 + 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

−10.03816682502883232848724505843, −9.140403206898220946961227682758, −8.261470901569093424534734056213, −7.53138311391475412234846459345, −6.94953164128169925072485600653, −5.68484105112531909640928874037, −4.47584796632881532587650291705, −3.86619415183519071335286697657, −2.80021793098482472301348633140, −0.837203931274100821703489157823, 0.837203931274100821703489157823, 2.80021793098482472301348633140, 3.86619415183519071335286697657, 4.47584796632881532587650291705, 5.68484105112531909640928874037, 6.94953164128169925072485600653, 7.53138311391475412234846459345, 8.261470901569093424534734056213, 9.140403206898220946961227682758, 10.03816682502883232848724505843

Graph of the ZZ-function along the critical line