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Properties

Label 4-249696-1.1-c1e2-0-3
Degree 44
Conductor 249696249696
Sign −1-1
Analytic cond. 15.920815.9208
Root an. cond. 1.997521.99752
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 8·11-s − 12-s − 4·13-s + 16-s − 18-s − 8·22-s + 24-s − 6·25-s + 4·26-s − 27-s − 32-s − 8·33-s + 36-s − 4·37-s + 4·39-s + 8·44-s − 48-s − 14·49-s + 6·50-s − 4·52-s + 54-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 2.41·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s − 1.70·22-s + 0.204·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s − 0.176·32-s − 1.39·33-s + 1/6·36-s − 0.657·37-s + 0.640·39-s + 1.20·44-s − 0.144·48-s − 2·49-s + 0.848·50-s − 0.554·52-s + 0.136·54-s + ⋯

Functional equation

Λ(s)=(249696s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(249696s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 249696249696    =    25⋅33⋅1722^{5} \cdot 3^{3} \cdot 17^{2}
Sign: −1-1
Analytic conductor: 15.920815.9208
Root analytic conductor: 1.997521.99752
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 249696, ( :1/2,1/2), −1)(4,\ 249696,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 1+T 1 + T
3C1C_1 1+T 1 + T
17C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.080746654144001493159861980940, −8.000937863938982294341146593643, −7.972624440390951709752181868622, −7.22142790948264379459326388401, −6.79533660929922109224513153399, −6.37869009161194820234692371426, −6.05886413503827459123865903257, −5.36244298721167160718725340278, −4.66119827871442481374467622682, −4.24541437163957543556604196065, −3.56656081546400251035965478912, −2.92323665285260202872977027719, −1.78155952274398559703453953653, −1.43438626728056627432143840048, 0, 1.43438626728056627432143840048, 1.78155952274398559703453953653, 2.92323665285260202872977027719, 3.56656081546400251035965478912, 4.24541437163957543556604196065, 4.66119827871442481374467622682, 5.36244298721167160718725340278, 6.05886413503827459123865903257, 6.37869009161194820234692371426, 6.79533660929922109224513153399, 7.22142790948264379459326388401, 7.972624440390951709752181868622, 8.000937863938982294341146593643, 9.080746654144001493159861980940

Graph of the ZZ-function along the critical line