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Properties

Label 4-648675-1.1-c1e2-0-2
Degree 44
Conductor 648675648675
Sign −1-1
Analytic cond. 41.360041.3600
Root an. cond. 2.535972.53597
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  − 3-s − 3·4-s + 9-s − 8·11-s + 3·12-s + 5·16-s + 4·17-s + 8·19-s + 25-s − 27-s − 4·29-s + 8·33-s − 3·36-s + 24·44-s − 5·48-s − 14·49-s − 4·51-s − 20·53-s − 8·57-s − 3·64-s + 24·67-s − 12·68-s − 75-s − 24·76-s + 81-s + 24·83-s + 4·87-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 1/3·9-s − 2.41·11-s + 0.866·12-s + 5/4·16-s + 0.970·17-s + 1.83·19-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.39·33-s − 1/2·36-s + 3.61·44-s − 0.721·48-s − 2·49-s − 0.560·51-s − 2.74·53-s − 1.05·57-s − 3/8·64-s + 2.93·67-s − 1.45·68-s − 0.115·75-s − 2.75·76-s + 1/9·81-s + 2.63·83-s + 0.428·87-s + ⋯

Functional equation

Λ(s)=(648675s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(648675s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 648675648675    =    33⋅52⋅3123^{3} \cdot 5^{2} \cdot 31^{2}
Sign: −1-1
Analytic conductor: 41.360041.3600
Root analytic conductor: 2.535972.53597
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 648675, ( :1/2,1/2), −1)(4,\ 648675,\ (\ :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)
bad3C1C_1 1+T 1 + T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
31C2C_2 1+pT2 1 + p T^{2}
good2C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + 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)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−8.114900408024103140014243113668, −7.66488013441745380243230842523, −7.62168634080583778803253039703, −6.76366529686096520107036215669, −6.22644927255442252578289500040, −5.48779277170674803346725706409, −5.23920392624592057055772361749, −5.11085823439540905400666440111, −4.62012066537284998425540488776, −3.83645868031632569816966802271, −3.25332827792826642629963089952, −2.91451080617330683387950783522, −1.86030328865315941936552250136, −0.850059814000557593825806016483, 0, 0.850059814000557593825806016483, 1.86030328865315941936552250136, 2.91451080617330683387950783522, 3.25332827792826642629963089952, 3.83645868031632569816966802271, 4.62012066537284998425540488776, 5.11085823439540905400666440111, 5.23920392624592057055772361749, 5.48779277170674803346725706409, 6.22644927255442252578289500040, 6.76366529686096520107036215669, 7.62168634080583778803253039703, 7.66488013441745380243230842523, 8.114900408024103140014243113668

Graph of the ZZ-function along the critical line