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Properties

Label 4-3960e2-1.1-c0e2-0-0
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 3.905753.90575
Root an. cond. 1.405801.40580
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 4·17-s − 25-s − 4·31-s − 6·32-s + 8·34-s − 2·49-s + 2·50-s + 8·62-s + 7·64-s − 12·68-s + 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s − 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 4·17-s − 25-s − 4·31-s − 6·32-s + 8·34-s − 2·49-s + 2·50-s + 8·62-s + 7·64-s − 12·68-s + 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s − 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 3.905753.90575
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15681600, ( :0,0), 1)(4,\ 15681600,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.054978602480.05497860248
L(12)L(\frac12) ≈\approx 0.054978602480.05497860248
L(1)L(1) 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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
3 1 1
5C2C_2 1+T2 1 + T^{2}
11C2C_2 1+T2 1 + T^{2}
good7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
17C1C_1 (1+T)4 ( 1 + T )^{4}
19C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C1C_1 (1+T)4 ( 1 + T )^{4}
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
43C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
47C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
59C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
79C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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

−9.004775370035875678549801517474, −8.509323935427188817897226767673, −8.209330854520343721483777713500, −7.82798336254782488132942299808, −7.37504485776600740668160493186, −6.99065378156788487976903773723, −6.90769472325328276005129820196, −6.40850687194899238472655216858, −6.16880070300042160950176763394, −5.61768290524021930345307389743, −5.32106622713297737144981824007, −4.61785614930289404693837308299, −4.24219269957321971810072800774, −3.49016413452305190273510091926, −3.47720801558350251072351154824, −2.38616257920987093853028129461, −2.37150468565054318804073876501, −1.81169405165641799571091610747, −1.53852408418074802007929967237, −0.16795327581745304143397884528, 0.16795327581745304143397884528, 1.53852408418074802007929967237, 1.81169405165641799571091610747, 2.37150468565054318804073876501, 2.38616257920987093853028129461, 3.47720801558350251072351154824, 3.49016413452305190273510091926, 4.24219269957321971810072800774, 4.61785614930289404693837308299, 5.32106622713297737144981824007, 5.61768290524021930345307389743, 6.16880070300042160950176763394, 6.40850687194899238472655216858, 6.90769472325328276005129820196, 6.99065378156788487976903773723, 7.37504485776600740668160493186, 7.82798336254782488132942299808, 8.209330854520343721483777713500, 8.509323935427188817897226767673, 9.004775370035875678549801517474

Graph of the ZZ-function along the critical line