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Properties

Label 4-1116-1.1-c1e2-0-0
Degree 44
Conductor 11161116
Sign 11
Analytic cond. 0.07115710.0711571
Root an. cond. 0.5164810.516481
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  − 3-s − 4-s + 7-s − 2·9-s + 12-s − 7·13-s + 16-s + 5·19-s − 21-s + 25-s + 5·27-s − 28-s + 8·31-s + 2·36-s + 37-s + 7·39-s − 2·43-s − 48-s − 7·49-s + 7·52-s − 5·57-s − 61-s − 2·63-s − 64-s − 14·67-s − 7·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 0.377·7-s − 2/3·9-s + 0.288·12-s − 1.94·13-s + 1/4·16-s + 1.14·19-s − 0.218·21-s + 1/5·25-s + 0.962·27-s − 0.188·28-s + 1.43·31-s + 1/3·36-s + 0.164·37-s + 1.12·39-s − 0.304·43-s − 0.144·48-s − 49-s + 0.970·52-s − 0.662·57-s − 0.128·61-s − 0.251·63-s − 1/8·64-s − 1.71·67-s − 0.819·73-s − 0.115·75-s + ⋯

Functional equation

Λ(s)=(1116s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1116s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 11161116    =    22⋅32⋅312^{2} \cdot 3^{2} \cdot 31
Sign: 11
Analytic conductor: 0.07115710.0711571
Root analytic conductor: 0.5164810.516481
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1116, ( :1/2,1/2), 1)(4,\ 1116,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.42542838170.4254283817
L(12)L(\frac12) ≈\approx 0.42542838170.4254283817
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)
bad2C2C_2 1+T2 1 + T^{2}
3C2C_2 1+T+pT2 1 + T + p T^{2}
31C1C_1×\timesC2C_2 (1−T)(1−7T+pT2) ( 1 - T )( 1 - 7 T + p T^{2} )
good5C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (1−3T+pT2)(1+2T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1+T+pT2)(1+6T+pT2) ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−5T+pT2)(1+pT2) ( 1 - 5 T + p T^{2} )( 1 + p T^{2} )
23C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
29C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−3T+pT2)(1+2T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−12T+pT2)(1+13T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} )
67C2C_2×\timesC2C_2 (1+2T+pT2)(1+12T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−4T+pT2)(1+11T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−5T+pT2)(1+pT2) ( 1 - 5 T + p T^{2} )( 1 + p T^{2} )
83C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
89C22C_2^2 1−72T2+p2T4 1 - 72 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−8T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{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

−14.08680482312613066827079930668, −13.57861727831866455994748310876, −12.70215991683975821083442252589, −11.94883798073866020533120603588, −11.80628704938984376814637709077, −10.89809460716251272126837437428, −10.06423424076045358628708764026, −9.599559494960929771349910383163, −8.718072445424649794509158417192, −7.899472857564502415835684557922, −7.19704267596062297832151605893, −6.12887186529772388269728794943, −5.15202294391188716329856472167, −4.65762024378348674735850325026, −2.92009081916655404042712155697, 2.92009081916655404042712155697, 4.65762024378348674735850325026, 5.15202294391188716329856472167, 6.12887186529772388269728794943, 7.19704267596062297832151605893, 7.899472857564502415835684557922, 8.718072445424649794509158417192, 9.599559494960929771349910383163, 10.06423424076045358628708764026, 10.89809460716251272126837437428, 11.80628704938984376814637709077, 11.94883798073866020533120603588, 12.70215991683975821083442252589, 13.57861727831866455994748310876, 14.08680482312613066827079930668

Graph of the ZZ-function along the critical line