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Properties

Label 4-60e3-1.1-c1e2-0-6
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 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 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 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 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

−8.986817345232143354160192898618, −8.624384126853811739476276912996, −8.222151557355997532915384866309, −7.63588507152931622264611878773, −6.99152317761459520818406697833, −6.66985734205751366789458017246, −6.44138367987963469752015935562, −5.57548968519602015653592828161, −5.08972778796573881835305963752, −4.21232854349724191593823438788, −3.85423103781006743254277566399, −3.48288543200374090452875304293, −2.70474550224093063124939423562, −1.89037268545330142266197998729, −0.893764133731049935751559998534, 0.893764133731049935751559998534, 1.89037268545330142266197998729, 2.70474550224093063124939423562, 3.48288543200374090452875304293, 3.85423103781006743254277566399, 4.21232854349724191593823438788, 5.08972778796573881835305963752, 5.57548968519602015653592828161, 6.44138367987963469752015935562, 6.66985734205751366789458017246, 6.99152317761459520818406697833, 7.63588507152931622264611878773, 8.222151557355997532915384866309, 8.624384126853811739476276912996, 8.986817345232143354160192898618

Graph of the ZZ-function along the critical line