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Properties

Label 4-3332e2-1.1-c0e2-0-7
Degree 44
Conductor 1110222411102224
Sign 11
Analytic cond. 2.765182.76518
Root an. cond. 1.289521.28952
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 − 2·9-s + 5·16-s − 4·18-s − 2·25-s + 6·32-s − 6·36-s − 4·50-s + 4·53-s + 7·64-s − 8·72-s + 3·81-s − 6·100-s + 8·106-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s − 2·25-s + 6·32-s − 6·36-s − 4·50-s + 4·53-s + 7·64-s − 8·72-s + 3·81-s − 6·100-s + 8·106-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + ⋯

Functional equation

Λ(s)=(11102224s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(11102224s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1110222411102224    =    24⋅74⋅1722^{4} \cdot 7^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 2.765182.76518
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11102224, ( :0,0), 1)(4,\ 11102224,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 4.9045591974.904559197
L(12)L(\frac12) ≈\approx 4.9045591974.904559197
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}
7 1 1
17C2C_2 1+T2 1 + T^{2}
good3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
11C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
19C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
23C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
37C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
41C2C_2 (1+T2)2 ( 1 + T^{2} )^{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}
53C1C_1 (1−T)4 ( 1 - T )^{4}
59C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
79C2C_2 (1+T2)2 ( 1 + T^{2} )^{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}
97C2C_2 (1+T2)2 ( 1 + T^{2} )^{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

−8.910798471797272718739241120261, −8.430633300389028200049488011712, −8.048066470631121885324039611231, −7.932166059349246756991103152183, −7.26929135940430910527311972030, −7.04115919702666259433507616079, −6.63573805565276938472088701074, −5.99821582395933511915893744867, −5.95268570667967206761893156753, −5.47196184114791683729290094088, −5.42316652431438731144289533028, −4.84989405626378519022573792706, −4.19642928662406139554867024405, −4.05546430443564320995967653251, −3.50207615625483771515222682221, −3.15546472061134942580040113142, −2.69363529614261304227035350493, −2.10603401765731893028781930880, −2.06773450539787846026591481102, −0.932916404743028582333596457171, 0.932916404743028582333596457171, 2.06773450539787846026591481102, 2.10603401765731893028781930880, 2.69363529614261304227035350493, 3.15546472061134942580040113142, 3.50207615625483771515222682221, 4.05546430443564320995967653251, 4.19642928662406139554867024405, 4.84989405626378519022573792706, 5.42316652431438731144289533028, 5.47196184114791683729290094088, 5.95268570667967206761893156753, 5.99821582395933511915893744867, 6.63573805565276938472088701074, 7.04115919702666259433507616079, 7.26929135940430910527311972030, 7.932166059349246756991103152183, 8.048066470631121885324039611231, 8.430633300389028200049488011712, 8.910798471797272718739241120261

Graph of the ZZ-function along the critical line