↑

Properties

Label 4-3332e2-1.1-c0e2-0-8
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 9-s + 2·13-s + 5·16-s + 2·17-s − 2·18-s + 2·25-s − 4·26-s − 6·32-s − 4·34-s + 3·36-s − 4·50-s + 6·52-s − 2·53-s + 7·64-s + 6·68-s − 4·72-s + 2·89-s + 6·100-s − 4·101-s − 8·104-s + 4·106-s + 2·117-s + 121-s + 127-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 9-s + 2·13-s + 5·16-s + 2·17-s − 2·18-s + 2·25-s − 4·26-s − 6·32-s − 4·34-s + 3·36-s − 4·50-s + 6·52-s − 2·53-s + 7·64-s + 6·68-s − 4·72-s + 2·89-s + 6·100-s − 4·101-s − 8·104-s + 4·106-s + 2·117-s + 121-s + 127-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 0.87658627640.8765862764
L(12)L(\frac12) ≈\approx 0.87658627640.8765862764
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
17C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
11C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
13C2C_2 (1−T+T2)2 ( 1 - T + 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}
41C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 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+T+T2)2 ( 1 + T + T^{2} )^{2}
59C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
61C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
67C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
71C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
show more
show less
   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

−8.973293562638251319724270053544, −8.620933075111954257645631536986, −8.200686561090330194341197503467, −8.056522561671168512024461503494, −7.68363495275501182808579406818, −7.10350550302749743541646205977, −7.03636006340168503341862007103, −6.52341598717714413264105181402, −6.07660500680430133180969592635, −5.98925987369782933819195683307, −5.31004788033584759314288198884, −4.98726622899125756363525248303, −4.21815871410334199376265175633, −3.64031054439535941364333605260, −3.29416024314644885509815301305, −3.00803724595188655270512892113, −2.34786515678965719531469512308, −1.54204151387104326027237222343, −1.28017951305091991985577267014, −0.960671531228963139956870116350, 0.960671531228963139956870116350, 1.28017951305091991985577267014, 1.54204151387104326027237222343, 2.34786515678965719531469512308, 3.00803724595188655270512892113, 3.29416024314644885509815301305, 3.64031054439535941364333605260, 4.21815871410334199376265175633, 4.98726622899125756363525248303, 5.31004788033584759314288198884, 5.98925987369782933819195683307, 6.07660500680430133180969592635, 6.52341598717714413264105181402, 7.03636006340168503341862007103, 7.10350550302749743541646205977, 7.68363495275501182808579406818, 8.056522561671168512024461503494, 8.200686561090330194341197503467, 8.620933075111954257645631536986, 8.973293562638251319724270053544

Graph of the ZZ-function along the critical line