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Properties

Label 4-665e2-1.1-c1e2-0-6
Degree 44
Conductor 442225442225
Sign 11
Analytic cond. 28.196628.1966
Root an. cond. 2.304352.30435
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s − 4-s − 2·5-s − 2·6-s − 4·7-s + 8·8-s + 3·9-s + 4·10-s + 5·11-s − 12-s + 3·13-s + 8·14-s − 2·15-s − 7·16-s − 5·17-s − 6·18-s + 8·19-s + 2·20-s − 4·21-s − 10·22-s − 9·23-s + 8·24-s + 3·25-s − 6·26-s + 8·27-s + 4·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.816·6-s − 1.51·7-s + 2.82·8-s + 9-s + 1.26·10-s + 1.50·11-s − 0.288·12-s + 0.832·13-s + 2.13·14-s − 0.516·15-s − 7/4·16-s − 1.21·17-s − 1.41·18-s + 1.83·19-s + 0.447·20-s − 0.872·21-s − 2.13·22-s − 1.87·23-s + 1.63·24-s + 3/5·25-s − 1.17·26-s + 1.53·27-s + 0.755·28-s + ⋯

Functional equation

Λ(s)=(442225s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(442225s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 442225442225    =    52⋅72⋅1925^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 28.196628.1966
Root analytic conductor: 2.304352.30435
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 442225, ( :1/2,1/2), 1)(4,\ 442225,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.71669923780.7166992378
L(12)L(\frac12) ≈\approx 0.71669923780.7166992378
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)
bad5C1C_1 (1+T)2 ( 1 + T )^{2}
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
19C2C_2 1−8T+pT2 1 - 8 T + p T^{2}
good2C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
3C22C_2^2 1−T−2T2−pT3+p2T4 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}
11C22C_2^2 1−5T+14T2−5pT3+p2T4 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−3T−4T2−3pT3+p2T4 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+5T+8T2+5pT3+p2T4 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+9T+58T2+9pT3+p2T4 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−9T+52T2−9pT3+p2T4 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+3T−22T2+3pT3+p2T4 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}
37C2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
41C22C_2^2 1−T−40T2−pT3+p2T4 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1+13T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} )
47C22C_2^2 1−11T+74T2−11pT3+p2T4 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−7T−10T2−7pT3+p2T4 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−5T−36T2−5pT3+p2T4 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C22C_2^2 1+7T−22T2+7pT3+p2T4 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+T−72T2+pT3+p2T4 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C22C_2^2 1−17T+200T2−17pT3+p2T4 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+13T+72T2+13pT3+p2T4 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4}
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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

−10.31201179122932016291704514637, −10.23240039682365054464786681347, −9.564495612730090921430754350442, −9.533204347436516416371844818571, −8.995411601444171612476030579584, −8.569823474406876398438481966605, −8.508056358228760756961154666897, −7.78250764314528178265085644923, −7.24997488309787050170000198673, −7.21025820957367086209681934264, −6.31962190650771400504595454890, −6.19068219209790629693712962741, −5.11316265743901630319528364038, −4.48888002085076148039709185438, −4.01257016378760112318340931330, −3.86485531249861752285261358268, −3.26137191542927263150707261882, −2.23388735747669767883354225607, −1.09941908342999494631582219071, −0.74357130591738392566940871635, 0.74357130591738392566940871635, 1.09941908342999494631582219071, 2.23388735747669767883354225607, 3.26137191542927263150707261882, 3.86485531249861752285261358268, 4.01257016378760112318340931330, 4.48888002085076148039709185438, 5.11316265743901630319528364038, 6.19068219209790629693712962741, 6.31962190650771400504595454890, 7.21025820957367086209681934264, 7.24997488309787050170000198673, 7.78250764314528178265085644923, 8.508056358228760756961154666897, 8.569823474406876398438481966605, 8.995411601444171612476030579584, 9.533204347436516416371844818571, 9.564495612730090921430754350442, 10.23240039682365054464786681347, 10.31201179122932016291704514637

Graph of the ZZ-function along the critical line