Properties

Label 4-170e2-1.1-c1e2-0-14
Degree 44
Conductor 2890028900
Sign 11
Analytic cond. 1.842681.84268
Root an. cond. 1.165091.16509
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 6·9-s − 4·13-s + 5·16-s − 2·17-s + 12·18-s − 16·19-s − 25-s − 8·26-s + 6·32-s − 4·34-s + 18·36-s − 32·38-s − 16·43-s + 10·49-s − 2·50-s − 12·52-s + 20·53-s + 8·59-s + 7·64-s − 8·67-s − 6·68-s + 24·72-s − 48·76-s + 27·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2·9-s − 1.10·13-s + 5/4·16-s − 0.485·17-s + 2.82·18-s − 3.67·19-s − 1/5·25-s − 1.56·26-s + 1.06·32-s − 0.685·34-s + 3·36-s − 5.19·38-s − 2.43·43-s + 10/7·49-s − 0.282·50-s − 1.66·52-s + 2.74·53-s + 1.04·59-s + 7/8·64-s − 0.977·67-s − 0.727·68-s + 2.82·72-s − 5.50·76-s + 3·81-s + ⋯

Functional equation

Λ(s)=(28900s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(28900s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2890028900    =    22521722^{2} \cdot 5^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 1.842681.84268
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 28900, ( :1/2,1/2), 1)(4,\ 28900,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.8785575792.878557579
L(12)L(\frac12) \approx 2.8785575792.878557579
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}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)Isogeny Class over Fp\mathbf{F}_p
bad2C1C_1 (1T)2 ( 1 - T )^{2}
5C2C_2 1+T2 1 + T^{2}
17C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good3C2C_2 (1pT2)2 ( 1 - p T^{2} )^{2} 2.3.a_ag
7C22C_2^2 110T2+p2T4 1 - 10 T^{2} + p^{2} T^{4} 2.7.a_ak
11C22C_2^2 16T2+p2T4 1 - 6 T^{2} + p^{2} T^{4} 2.11.a_ag
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2} 2.13.e_be
19C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2} 2.19.q_dy
23C22C_2^2 110T2+p2T4 1 - 10 T^{2} + p^{2} T^{4} 2.23.a_ak
29C22C_2^2 122T2+p2T4 1 - 22 T^{2} + p^{2} T^{4} 2.29.a_aw
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4} 2.31.a_bm
37C2C_2 (112T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) 2.37.a_acs
41C22C_2^2 166T2+p2T4 1 - 66 T^{2} + p^{2} T^{4} 2.41.a_aco
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2} 2.43.q_fu
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.47.a_dq
53C2C_2 (110T+pT2)2 ( 1 - 10 T + p T^{2} )^{2} 2.53.au_hy
59C2C_2 (14T+pT2)2 ( 1 - 4 T + p T^{2} )^{2} 2.59.ai_fe
61C2C_2 (112T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) 2.61.a_aw
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2} 2.67.i_fu
71C22C_2^2 142T2+p2T4 1 - 42 T^{2} + p^{2} T^{4} 2.71.a_abq
73C22C_2^2 12T2+p2T4 1 - 2 T^{2} + p^{2} T^{4} 2.73.a_ac
79C22C_2^2 1122T2+p2T4 1 - 122 T^{2} + p^{2} T^{4} 2.79.a_aes
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.83.a_gk
89C2C_2 (12T+pT2)2 ( 1 - 2 T + p T^{2} )^{2} 2.89.ae_ha
97C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4} 2.97.a_ck
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   L(s)=p j=14(1αj,pps)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

−12.94407642036537484328674743374, −12.71991391950053845994315374925, −12.19336794839424565471569783485, −11.85433668855291833341100897616, −11.05543305765230238128605601836, −10.55622503560228451325784433348, −10.11639100699523557853309705068, −9.945609295567579133963362296170, −8.719628844099429559617781278284, −8.538087315103129226148980569814, −7.37919842515154954497070386172, −7.26835572617531933661978272572, −6.45168467505020539170819963411, −6.34769481191182450507979128760, −5.21583820190124830450662575045, −4.66481553244488061897850650772, −4.16309584125763120004185199517, −3.81144439505337770209116126257, −2.35577095351504871124232407459, −1.96873713218844423806568882524, 1.96873713218844423806568882524, 2.35577095351504871124232407459, 3.81144439505337770209116126257, 4.16309584125763120004185199517, 4.66481553244488061897850650772, 5.21583820190124830450662575045, 6.34769481191182450507979128760, 6.45168467505020539170819963411, 7.26835572617531933661978272572, 7.37919842515154954497070386172, 8.538087315103129226148980569814, 8.719628844099429559617781278284, 9.945609295567579133963362296170, 10.11639100699523557853309705068, 10.55622503560228451325784433348, 11.05543305765230238128605601836, 11.85433668855291833341100897616, 12.19336794839424565471569783485, 12.71991391950053845994315374925, 12.94407642036537484328674743374

Graph of the ZZ-function along the critical line