Properties

Label 2-4998-1.1-c1-0-57
Degree 22
Conductor 49984998
Sign 11
Analytic cond. 39.909239.9092
Root an. cond. 6.317376.31737
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-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 12-s + 6·13-s + 4·15-s + 16-s + 17-s − 18-s − 4·19-s + 4·20-s + 6·23-s − 24-s + 11·25-s − 6·26-s + 27-s − 4·29-s − 4·30-s + 6·31-s − 32-s − 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s + 1.66·13-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.894·20-s + 1.25·23-s − 0.204·24-s + 11/5·25-s − 1.17·26-s + 0.192·27-s − 0.742·29-s − 0.730·30-s + 1.07·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + ⋯

Functional equation

Λ(s)=(4998s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4998s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49984998    =    2372172 \cdot 3 \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 39.909239.9092
Root analytic conductor: 6.317376.31737
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4998, ( :1/2), 1)(2,\ 4998,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.9448838322.944883832
L(12)L(\frac12) \approx 2.9448838322.944883832
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+T 1 + T
3 1T 1 - T
7 1 1
17 1T 1 - T
good5 14T+pT2 1 - 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 16T+pT2 1 - 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 16T+pT2 1 - 6 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 16T+pT2 1 - 6 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 110T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 14T+pT2 1 - 4 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 110T+pT2 1 - 10 T + p T^{2}
83 112T+pT2 1 - 12 T + p T^{2}
89 12T+pT2 1 - 2 T + p T^{2}
97 1+6T+pT2 1 + 6 T + p T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.511277482343380670875082616936, −7.67080076534670349964439281106, −6.65609487669640152451998833504, −6.25650993618347018284029221741, −5.58930669699156824874707798240, −4.60406618583419097735195221715, −3.41650850636158117457558100754, −2.65274646328445039923916322365, −1.75668697624836907999103677619, −1.13374758634790644463533329227, 1.13374758634790644463533329227, 1.75668697624836907999103677619, 2.65274646328445039923916322365, 3.41650850636158117457558100754, 4.60406618583419097735195221715, 5.58930669699156824874707798240, 6.25650993618347018284029221741, 6.65609487669640152451998833504, 7.67080076534670349964439281106, 8.511277482343380670875082616936

Graph of the ZZ-function along the critical line