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Properties

Label 4-525e2-1.1-c3e2-0-6
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 959.512959.512
Root an. cond. 5.565605.56560
Motivic weight 33
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  + 3·2-s − 6·3-s − 5·4-s − 18·6-s − 14·7-s − 33·8-s + 27·9-s − 31·11-s + 30·12-s + 39·13-s − 42·14-s − 21·16-s − 79·17-s + 81·18-s − 56·19-s + 84·21-s − 93·22-s + 254·23-s + 198·24-s + 117·26-s − 108·27-s + 70·28-s − 62·29-s − 135·31-s + 87·32-s + 186·33-s − 237·34-s + ⋯
L(s)  = 1  + 1.06·2-s − 1.15·3-s − 5/8·4-s − 1.22·6-s − 0.755·7-s − 1.45·8-s + 9-s − 0.849·11-s + 0.721·12-s + 0.832·13-s − 0.801·14-s − 0.328·16-s − 1.12·17-s + 1.06·18-s − 0.676·19-s + 0.872·21-s − 0.901·22-s + 2.30·23-s + 1.68·24-s + 0.882·26-s − 0.769·27-s + 0.472·28-s − 0.397·29-s − 0.782·31-s + 0.480·32-s + 0.981·33-s − 1.19·34-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 959.512959.512
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :3/2,3/2), 1)(4,\ 275625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4254738761.425473876
L(12)L(\frac12) ≈\approx 1.4254738761.425473876
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)
bad3C1C_1 (1+pT)2 ( 1 + p T )^{2}
5 1 1
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2C4C_4 1−3T+7pT2−3p3T3+p6T4 1 - 3 T + 7 p T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+31T+2796T2+31p3T3+p6T4 1 + 31 T + 2796 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−3pT+3546T2−3p4T3+p6T4 1 - 3 p T + 3546 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4}
17D4D_{4} 1+79T+8288T2+79p3T3+p6T4 1 + 79 T + 8288 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+56T+1174T2+56p3T3+p6T4 1 + 56 T + 1174 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−254T+38015T2−254p3T3+p6T4 1 - 254 T + 38015 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+62T+19751T2+62p3T3+p6T4 1 + 62 T + 19751 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+135T+63420T2+135p3T3+p6T4 1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−113T+102964T2−113p3T3+p6T4 1 - 113 T + 102964 T^{2} - 113 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−235T+143042T2−235p3T3+p6T4 1 - 235 T + 143042 T^{2} - 235 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−804T+306321T2−804p3T3+p6T4 1 - 804 T + 306321 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−152T+180510T2−152p3T3+p6T4 1 - 152 T + 180510 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−149T+269640T2−149p3T3+p6T4 1 - 149 T + 269640 T^{2} - 149 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+441T+273734T2+441p3T3+p6T4 1 + 441 T + 273734 T^{2} + 441 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+223T+149646T2+223p3T3+p6T4 1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−1157T+871890T2−1157p3T3+p6T4 1 - 1157 T + 871890 T^{2} - 1157 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−619T+576906T2−619p3T3+p6T4 1 - 619 T + 576906 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−268T+763078T2−268p3T3+p6T4 1 - 268 T + 763078 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+427T+986572T2+427p3T3+p6T4 1 + 427 T + 986572 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−1211T+720720T2−1211p3T3+p6T4 1 - 1211 T + 720720 T^{2} - 1211 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−466T+306170T2−466p3T3+p6T4 1 - 466 T + 306170 T^{2} - 466 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−172T−273626T2−172p3T3+p6T4 1 - 172 T - 273626 T^{2} - 172 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

−10.85919753065442622498907170085, −10.59333444576037829179880456064, −9.601977875832480089579576227904, −9.468960479219499458394271749213, −8.849056421123131864142375321339, −8.778351498589352675205054411990, −7.81572400732461855890984495501, −7.39551954694108854298745981874, −6.77388928272800678244653833131, −6.40980288269180802578694535687, −5.72902192319760416974702302077, −5.67587338633174292972238141469, −4.85052483572484563262538781795, −4.75123181023720967039259971988, −3.93060693162802408963263995972, −3.77827571140795938008556959803, −2.86327279723027258438942442193, −2.25968941286165932325369206806, −0.929427952112897476085075556694, −0.43840591368660462215669846199, 0.43840591368660462215669846199, 0.929427952112897476085075556694, 2.25968941286165932325369206806, 2.86327279723027258438942442193, 3.77827571140795938008556959803, 3.93060693162802408963263995972, 4.75123181023720967039259971988, 4.85052483572484563262538781795, 5.67587338633174292972238141469, 5.72902192319760416974702302077, 6.40980288269180802578694535687, 6.77388928272800678244653833131, 7.39551954694108854298745981874, 7.81572400732461855890984495501, 8.778351498589352675205054411990, 8.849056421123131864142375321339, 9.468960479219499458394271749213, 9.601977875832480089579576227904, 10.59333444576037829179880456064, 10.85919753065442622498907170085

Graph of the ZZ-function along the critical line