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Properties

Label 2-936-1.1-c1-0-8
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  + 2·5-s + 4·7-s + 13-s − 2·17-s + 8·19-s − 8·23-s − 25-s + 2·29-s + 4·31-s + 8·35-s − 10·37-s − 2·41-s − 4·43-s + 12·47-s + 9·49-s − 6·53-s − 2·61-s + 2·65-s + 8·67-s + 12·71-s + 10·73-s − 8·79-s − 4·85-s + 14·89-s + 4·91-s + 16·95-s + 2·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s + 0.277·13-s − 0.485·17-s + 1.83·19-s − 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 1.35·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.256·61-s + 0.248·65-s + 0.977·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s − 0.433·85-s + 1.48·89-s + 0.419·91-s + 1.64·95-s + 0.203·97-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 2.2013342172.201334217
L(12)L(\frac12) ≈\approx 2.2013342172.201334217
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 1−2T+pT2 1 - 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+pT2 1 + 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+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+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+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−14T+pT2 1 - 14 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

−10.06728738330013813743566474530, −9.279202530789602619833830982926, −8.305939490605595897046504729212, −7.70165993250910980680050902027, −6.58583463773293315964362259330, −5.55956767485975536855936647512, −4.97706613598293520985338882848, −3.79711754844167209811925133514, −2.30100750298902383089718144360, −1.38372329984625885819535224184, 1.38372329984625885819535224184, 2.30100750298902383089718144360, 3.79711754844167209811925133514, 4.97706613598293520985338882848, 5.55956767485975536855936647512, 6.58583463773293315964362259330, 7.70165993250910980680050902027, 8.305939490605595897046504729212, 9.279202530789602619833830982926, 10.06728738330013813743566474530

Graph of the ZZ-function along the critical line