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Properties

Label 4-2205e2-1.1-c3e2-0-12
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  + 6·2-s + 13·4-s + 10·5-s − 12·8-s + 60·10-s − 28·11-s + 36·13-s − 147·16-s − 76·17-s − 160·19-s + 130·20-s − 168·22-s + 22·23-s + 75·25-s + 216·26-s + 250·29-s + 132·31-s − 366·32-s − 456·34-s − 416·37-s − 960·38-s − 120·40-s − 106·41-s − 666·43-s − 364·44-s + 132·46-s − 196·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 13/8·4-s + 0.894·5-s − 0.530·8-s + 1.89·10-s − 0.767·11-s + 0.768·13-s − 2.29·16-s − 1.08·17-s − 1.93·19-s + 1.45·20-s − 1.62·22-s + 0.199·23-s + 3/5·25-s + 1.62·26-s + 1.60·29-s + 0.764·31-s − 2.02·32-s − 2.30·34-s − 1.84·37-s − 4.09·38-s − 0.474·40-s − 0.403·41-s − 2.36·43-s − 1.24·44-s + 0.423·46-s − 0.608·47-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} 1−3pT+23T2−3p4T3+p6T4 1 - 3 p T + 23 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4}
11D4D_{4} 1+28T+2658T2+28p3T3+p6T4 1 + 28 T + 2658 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−36T+4518T2−36p3T3+p6T4 1 - 36 T + 4518 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+76T+5438T2+76p3T3+p6T4 1 + 76 T + 5438 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+160T+18766T2+160p3T3+p6T4 1 + 160 T + 18766 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−22T−2923T2−22p3T3+p6T4 1 - 22 T - 2923 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−250T+57203T2−250p3T3+p6T4 1 - 250 T + 57203 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−132T+43938T2−132p3T3+p6T4 1 - 132 T + 43938 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+416T+107578T2+416p3T3+p6T4 1 + 416 T + 107578 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+106T+138851T2+106p3T3+p6T4 1 + 106 T + 138851 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+666T+269853T2+666p3T3+p6T4 1 + 666 T + 269853 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+196T+209058T2+196p3T3+p6T4 1 + 196 T + 209058 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−952T+484002T2−952p3T3+p6T4 1 - 952 T + 484002 T^{2} - 952 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−840T+445646T2−840p3T3+p6T4 1 - 840 T + 445646 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+98T−193437T2+98p3T3+p6T4 1 + 98 T - 193437 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+1286T+1005453T2+1286p3T3+p6T4 1 + 1286 T + 1005453 T^{2} + 1286 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+1064T+753846T2+1064p3T3+p6T4 1 + 1064 T + 753846 T^{2} + 1064 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−172T+757582T2−172p3T3+p6T4 1 - 172 T + 757582 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+1240T+1278886T2+1240p3T3+p6T4 1 + 1240 T + 1278886 T^{2} + 1240 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1906T+2051733T2+1906p3T3+p6T4 1 + 1906 T + 2051733 T^{2} + 1906 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−650T+1305611T2−650p3T3+p6T4 1 - 650 T + 1305611 T^{2} - 650 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−628T+1423942T2−628p3T3+p6T4 1 - 628 T + 1423942 T^{2} - 628 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

−8.636507046398640773636141976387, −8.453565340003411716840278633476, −7.54886544649725310425440671097, −7.02359095070133088163426710018, −6.57393217042474465696054973490, −6.49268043773588287301138132624, −5.89444212664407642106808594454, −5.79163956650594377421196551775, −5.11673505555507812672986026592, −4.88877306990796603936891745585, −4.56161651225160412598496109888, −4.18632835847452375611412491895, −3.64759922069914234253102412208, −3.27974395025858111346575102147, −2.63423560755013275398142139247, −2.46860401114981664092656161682, −1.79423959066463284730612155503, −1.20295674488558415113867060658, 0, 0, 1.20295674488558415113867060658, 1.79423959066463284730612155503, 2.46860401114981664092656161682, 2.63423560755013275398142139247, 3.27974395025858111346575102147, 3.64759922069914234253102412208, 4.18632835847452375611412491895, 4.56161651225160412598496109888, 4.88877306990796603936891745585, 5.11673505555507812672986026592, 5.79163956650594377421196551775, 5.89444212664407642106808594454, 6.49268043773588287301138132624, 6.57393217042474465696054973490, 7.02359095070133088163426710018, 7.54886544649725310425440671097, 8.453565340003411716840278633476, 8.636507046398640773636141976387

Graph of the ZZ-function along the critical line