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Properties

Label 2-99705-1.1-c1-0-5
Degree 22
Conductor 9970599705
Sign 11
Analytic cond. 796.148796.148
Root an. cond. 28.216128.2161
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  − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 5·7-s + 9-s + 2·10-s + 2·11-s − 2·12-s − 6·13-s − 10·14-s + 15-s − 4·16-s − 2·18-s + 2·19-s − 2·20-s − 5·21-s − 4·22-s + 23-s + 25-s + 12·26-s − 27-s + 10·28-s + 29-s − 2·30-s + 5·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1.88·7-s + 1/3·9-s + 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.66·13-s − 2.67·14-s + 0.258·15-s − 16-s − 0.471·18-s + 0.458·19-s − 0.447·20-s − 1.09·21-s − 0.852·22-s + 0.208·23-s + 1/5·25-s + 2.35·26-s − 0.192·27-s + 1.88·28-s + 0.185·29-s − 0.365·30-s + 0.898·31-s + ⋯

Functional equation

Λ(s)=(99705s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 99705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(99705s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9970599705    =    3⋅5⋅172⋅233 \cdot 5 \cdot 17^{2} \cdot 23
Sign: 11
Analytic conductor: 796.148796.148
Root analytic conductor: 28.216128.2161
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 99705, ( :1/2), 1)(2,\ 99705,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0154777801.015477780
L(12)L(\frac12) ≈\approx 1.0154777801.015477780
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)
bad3 1+T 1 + T
5 1+T 1 + T
17 1 1
23 1−T 1 - T
good2 1+pT+pT2 1 + p T + p T^{2}
7 1−5T+pT2 1 - 5 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−7T+pT2 1 - 7 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1+T+pT2 1 + T + p T^{2}
71 1+5T+pT2 1 + 5 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 1−3T+pT2 1 - 3 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−13.71561571096326, −13.39780831262481, −12.39478816142619, −12.03012157105492, −11.65040658235921, −11.25842471335840, −10.83166432317135, −10.28601946410999, −9.752623361010487, −9.377595446064628, −8.731561331377690, −8.192613757508338, −7.786630358631465, −7.508893226062053, −6.898283642786007, −6.417456417398038, −5.444959133101965, −5.012775214793285, −4.458441690624613, −4.229238674310422, −3.005462343560139, −2.294672729614163, −1.676773913590966, −1.099146571028098, −0.4862772367046520, 0.4862772367046520, 1.099146571028098, 1.676773913590966, 2.294672729614163, 3.005462343560139, 4.229238674310422, 4.458441690624613, 5.012775214793285, 5.444959133101965, 6.417456417398038, 6.898283642786007, 7.508893226062053, 7.786630358631465, 8.192613757508338, 8.731561331377690, 9.377595446064628, 9.752623361010487, 10.28601946410999, 10.83166432317135, 11.25842471335840, 11.65040658235921, 12.03012157105492, 12.39478816142619, 13.39780831262481, 13.71561571096326

Graph of the ZZ-function along the critical line