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Properties

Label 2-3520-1.1-c1-0-19
Degree 22
Conductor 35203520
Sign 11
Analytic cond. 28.107328.1073
Root an. cond. 5.301635.30163
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  + 2·3-s − 5-s − 4·7-s + 9-s − 11-s − 4·13-s − 2·15-s + 4·17-s + 4·19-s − 8·21-s + 6·23-s + 25-s − 4·27-s + 10·29-s + 4·31-s − 2·33-s + 4·35-s − 2·37-s − 8·39-s − 10·41-s + 8·43-s − 45-s − 6·47-s + 9·49-s + 8·51-s − 2·53-s + 55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.516·15-s + 0.970·17-s + 0.917·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s + 0.718·31-s − 0.348·33-s + 0.676·35-s − 0.328·37-s − 1.28·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 1.12·51-s − 0.274·53-s + 0.134·55-s + ⋯

Functional equation

Λ(s)=(3520s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3520s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35203520    =    26⋅5⋅112^{6} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 28.107328.1073
Root analytic conductor: 5.301635.30163
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3520, ( :1/2), 1)(2,\ 3520,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9300066481.930006648
L(12)L(\frac12) ≈\approx 1.9300066481.930006648
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
5 1+T 1 + T
11 1+T 1 + T
good3 1−2T+pT2 1 - 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−8T+pT2 1 - 8 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 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−12T+pT2 1 - 12 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+10T+pT2 1 + 10 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

−8.527521801955114252432580447305, −7.923580520667090509256462573465, −7.14162936849208426215495288495, −6.62536331191924721842679282461, −5.48691634933986413556576113146, −4.69737474865266608096337291625, −3.42894015870646694064785929471, −3.15044879752992676393154963603, −2.43460898771104914069889479972, −0.75069098874977159462101565047, 0.75069098874977159462101565047, 2.43460898771104914069889479972, 3.15044879752992676393154963603, 3.42894015870646694064785929471, 4.69737474865266608096337291625, 5.48691634933986413556576113146, 6.62536331191924721842679282461, 7.14162936849208426215495288495, 7.923580520667090509256462573465, 8.527521801955114252432580447305

Graph of the ZZ-function along the critical line