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Properties

Label 4-2205e2-1.1-c3e2-0-8
Degree 44
Conductor 48620254862025
Sign 11
Analytic cond. 16925.816925.8
Root an. cond. 11.406111.4061
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 11·4-s − 10·5-s − 15·8-s − 10·10-s − 4·11-s + 22·13-s + 61·16-s + 58·17-s + 110·20-s − 4·22-s − 82·23-s + 75·25-s + 22·26-s − 334·29-s + 210·31-s + 89·32-s + 58·34-s + 6·37-s + 150·40-s + 176·41-s + 46·43-s + 44·44-s − 82·46-s + 514·47-s + 75·50-s − 242·52-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.37·4-s − 0.894·5-s − 0.662·8-s − 0.316·10-s − 0.109·11-s + 0.469·13-s + 0.953·16-s + 0.827·17-s + 1.22·20-s − 0.0387·22-s − 0.743·23-s + 3/5·25-s + 0.165·26-s − 2.13·29-s + 1.21·31-s + 0.491·32-s + 0.292·34-s + 0.0266·37-s + 0.592·40-s + 0.670·41-s + 0.163·43-s + 0.150·44-s − 0.262·46-s + 1.59·47-s + 0.212·50-s − 0.645·52-s + ⋯

Functional equation

Λ(s)=(4862025s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(4862025s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 48620254862025    =    34⋅52⋅743^{4} \cdot 5^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 16925.816925.8
Root analytic conductor: 11.406111.4061
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 4862025, ( :3/2,3/2), 1)(4,\ 4862025,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
5C1C_1 (1+pT)2 ( 1 + p T )^{2}
7 1 1
good2D4D_{4} 1−T+3p2T2−p3T3+p6T4 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+4T+2598T2+4p3T3+p6T4 1 + 4 T + 2598 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−22T+2458T2−22p3T3+p6T4 1 - 22 T + 2458 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−58T+7794T2−58p3T3+p6T4 1 - 58 T + 7794 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1+5490T2+p6T4 1 + 5490 T^{2} + p^{6} T^{4}
23D4D_{4} 1+82T+25182T2+82p3T3+p6T4 1 + 82 T + 25182 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+334T+72842T2+334p3T3+p6T4 1 + 334 T + 72842 T^{2} + 334 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−210T+69230T2−210p3T3+p6T4 1 - 210 T + 69230 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−6T+97490T2−6p3T3+p6T4 1 - 6 T + 97490 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−176T+134094T2−176p3T3+p6T4 1 - 176 T + 134094 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−46T+42430T2−46p3T3+p6T4 1 - 46 T + 42430 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−514T+269870T2−514p3T3+p6T4 1 - 514 T + 269870 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+808T+449478T2+808p3T3+p6T4 1 + 808 T + 449478 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4}
59C2C_2 (1−284T+p3T2)2 ( 1 - 284 T + p^{3} T^{2} )^{2}
61D4D_{4} 1−618T+515018T2−618p3T3+p6T4 1 - 618 T + 515018 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−694T+681118T2−694p3T3+p6T4 1 - 694 T + 681118 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+814T+769934T2+814p3T3+p6T4 1 + 814 T + 769934 T^{2} + 814 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+82T+422290T2+82p3T3+p6T4 1 + 82 T + 422290 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−600T+618846T2−600p3T3+p6T4 1 - 600 T + 618846 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+268T+779030T2+268p3T3+p6T4 1 + 268 T + 779030 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+72T+448286T2+72p3T3+p6T4 1 + 72 T + 448286 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1626T+1498938T2+1626p3T3+p6T4 1 + 1626 T + 1498938 T^{2} + 1626 p^{3} T^{3} + p^{6} 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

−8.313986404571865142652630448011, −8.282109135128132622762307548727, −7.75866819823711929078405531565, −7.52195641179090705481838590446, −7.02953864171686226132912444075, −6.52207133860112242905809928598, −5.89119935430835704372433845551, −5.79689757782884252790490275939, −5.11019922436989186607765222217, −5.01786645271672186444496826923, −4.29555929638007642409199067416, −4.03945007081128344518315577254, −3.69724288265100475036162569025, −3.41955156059658656429943430252, −2.66483754607630004393587062209, −2.20903851023028807854969532293, −1.23956327088675683827205019265, −0.944177348487544557330072115825, 0, 0, 0.944177348487544557330072115825, 1.23956327088675683827205019265, 2.20903851023028807854969532293, 2.66483754607630004393587062209, 3.41955156059658656429943430252, 3.69724288265100475036162569025, 4.03945007081128344518315577254, 4.29555929638007642409199067416, 5.01786645271672186444496826923, 5.11019922436989186607765222217, 5.79689757782884252790490275939, 5.89119935430835704372433845551, 6.52207133860112242905809928598, 7.02953864171686226132912444075, 7.52195641179090705481838590446, 7.75866819823711929078405531565, 8.282109135128132622762307548727, 8.313986404571865142652630448011

Graph of the ZZ-function along the critical line