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Properties

Label 4-60e3-1.1-c1e2-0-10
Degree 44
Conductor 216000216000
Sign 11
Analytic cond. 13.772313.7723
Root an. cond. 1.926421.92642
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 + 5-s + 2·7-s + 9-s + 6·11-s − 15-s − 2·21-s + 25-s − 27-s − 6·33-s + 2·35-s + 8·43-s + 45-s − 2·49-s − 12·53-s + 6·55-s + 18·59-s + 4·61-s + 2·63-s − 4·67-s + 12·71-s − 75-s + 12·77-s + 81-s + 6·99-s − 10·103-s − 2·105-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.258·15-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.04·33-s + 0.338·35-s + 1.21·43-s + 0.149·45-s − 2/7·49-s − 1.64·53-s + 0.809·55-s + 2.34·59-s + 0.512·61-s + 0.251·63-s − 0.488·67-s + 1.42·71-s − 0.115·75-s + 1.36·77-s + 1/9·81-s + 0.603·99-s − 0.985·103-s − 0.195·105-s + ⋯

Functional equation

Λ(s)=(216000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(216000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 216000216000    =    26⋅33⋅532^{6} \cdot 3^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 13.772313.7723
Root analytic conductor: 1.926421.92642
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 216000, ( :1/2,1/2), 1)(4,\ 216000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9639078071.963907807
L(12)L(\frac12) ≈\approx 1.9639078071.963907807
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)
bad2 1 1
3C1C_1 1+T 1 + T
5C1C_1 1−T 1 - T
good7C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
13C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2×\timesC2C_2 (1−12T+pT2)(1−6T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1+pT2) ( 1 - 12 T + p T^{2} )( 1 + p T^{2} )
73C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
89C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
97C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 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

−9.141241154029135032259294624053, −8.609298928202332924153573545631, −8.111123297949343578743720333764, −7.62278521410384067925608890140, −6.93159405149276696220808232612, −6.63621551256985941580492577404, −6.20367239541730683512107851461, −5.56354189044085788017222900632, −5.17790188564623764239537996870, −4.47799600643106816674367720352, −4.05726476535751679542914610079, −3.46610539182281159391386066816, −2.46858524619923688484799519620, −1.68298604675044022991253635007, −1.01603419970345686879799970894, 1.01603419970345686879799970894, 1.68298604675044022991253635007, 2.46858524619923688484799519620, 3.46610539182281159391386066816, 4.05726476535751679542914610079, 4.47799600643106816674367720352, 5.17790188564623764239537996870, 5.56354189044085788017222900632, 6.20367239541730683512107851461, 6.63621551256985941580492577404, 6.93159405149276696220808232612, 7.62278521410384067925608890140, 8.111123297949343578743720333764, 8.609298928202332924153573545631, 9.141241154029135032259294624053

Graph of the ZZ-function along the critical line