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Properties

Label 4-136e2-1.1-c0e2-0-0
Degree 44
Conductor 1849618496
Sign 11
Analytic cond. 0.004606720.00460672
Root an. cond. 0.2605240.260524
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  − 2·3-s − 4-s + 2·9-s + 2·11-s + 2·12-s + 16-s − 2·27-s − 4·33-s − 2·36-s − 2·41-s − 2·44-s − 2·48-s − 64-s − 2·73-s + 3·81-s + 2·97-s + 4·99-s + 2·107-s + 2·108-s − 2·113-s + 2·121-s + 4·123-s + 127-s + 131-s + 4·132-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·3-s − 4-s + 2·9-s + 2·11-s + 2·12-s + 16-s − 2·27-s − 4·33-s − 2·36-s − 2·41-s − 2·44-s − 2·48-s − 64-s − 2·73-s + 3·81-s + 2·97-s + 4·99-s + 2·107-s + 2·108-s − 2·113-s + 2·121-s + 4·123-s + 127-s + 131-s + 4·132-s + 137-s + 139-s + ⋯

Functional equation

Λ(s)=(18496s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(18496s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1849618496    =    26⋅1722^{6} \cdot 17^{2}
Sign: 11
Analytic conductor: 0.004606720.00460672
Root analytic conductor: 0.2605240.260524
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 18496, ( :0,0), 1)(4,\ 18496,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.20246120970.2024612097
L(12)L(\frac12) ≈\approx 0.20246120970.2024612097
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)
bad2C2C_2 1+T2 1 + T^{2}
17C2C_2 1+T2 1 + T^{2}
good3C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
5C22C_2^2 1+T4 1 + T^{4}
7C22C_2^2 1+T4 1 + T^{4}
11C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
13C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
23C22C_2^2 1+T4 1 + T^{4}
29C22C_2^2 1+T4 1 + T^{4}
31C22C_2^2 1+T4 1 + T^{4}
37C22C_2^2 1+T4 1 + T^{4}
41C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
43C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{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}
59C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
61C22C_2^2 1+T4 1 + T^{4}
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C22C_2^2 1+T4 1 + T^{4}
73C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
79C22C_2^2 1+T4 1 + T^{4}
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×\timesC2C_2 (1−T)2(1+T2) ( 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

−14.10829379018122111854593772233, −12.85715835520574200022236771097, −12.83260234817688541120987881577, −11.77449855381196807410731862227, −11.76561960022422668253953730470, −11.58875341989955402464879387225, −10.57382924601604411035752812235, −10.31820460079360129722653774897, −9.663511030670705522782666378170, −9.083110844705150311940738088582, −8.718488301401035354803204082389, −7.80649168776885473376423669888, −7.09243364214847237611653603262, −6.35973643123559432573863254683, −6.15309186342952232387332769817, −5.33625654831850659557344740679, −4.89202963436691088873700380710, −4.12555615364533978441904967912, −3.55644053736731572668032407830, −1.42085106340463447177544392325, 1.42085106340463447177544392325, 3.55644053736731572668032407830, 4.12555615364533978441904967912, 4.89202963436691088873700380710, 5.33625654831850659557344740679, 6.15309186342952232387332769817, 6.35973643123559432573863254683, 7.09243364214847237611653603262, 7.80649168776885473376423669888, 8.718488301401035354803204082389, 9.083110844705150311940738088582, 9.663511030670705522782666378170, 10.31820460079360129722653774897, 10.57382924601604411035752812235, 11.58875341989955402464879387225, 11.76561960022422668253953730470, 11.77449855381196807410731862227, 12.83260234817688541120987881577, 12.85715835520574200022236771097, 14.10829379018122111854593772233

Graph of the ZZ-function along the critical line