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Properties

Label 4-150e2-1.1-c9e2-0-11
Degree 44
Conductor 2250022500
Sign 11
Analytic cond. 5968.395968.39
Root an. cond. 8.789508.78950
Motivic weight 99
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  − 32·2-s − 162·3-s + 768·4-s + 5.18e3·6-s − 3.16e3·7-s − 1.63e4·8-s + 1.96e4·9-s + 2.70e4·11-s − 1.24e5·12-s − 7.08e4·13-s + 1.01e5·14-s + 3.27e5·16-s − 1.44e5·17-s − 6.29e5·18-s + 1.15e5·19-s + 5.12e5·21-s − 8.66e5·22-s − 7.99e5·23-s + 2.65e6·24-s + 2.26e6·26-s − 2.12e6·27-s − 2.43e6·28-s − 2.08e5·29-s + 6.36e6·31-s − 6.29e6·32-s − 4.38e6·33-s + 4.63e6·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.498·7-s − 1.41·8-s + 9-s + 0.557·11-s − 1.73·12-s − 0.688·13-s + 0.704·14-s + 5/4·16-s − 0.420·17-s − 1.41·18-s + 0.203·19-s + 0.575·21-s − 0.788·22-s − 0.595·23-s + 1.63·24-s + 0.973·26-s − 0.769·27-s − 0.747·28-s − 0.0546·29-s + 1.23·31-s − 1.06·32-s − 0.644·33-s + 0.594·34-s + ⋯

Functional equation

Λ(s)=(22500s/2 ΓC(s)2 L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(22500s/2 ΓC(s+9/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2250022500    =    22⋅32⋅542^{2} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 5968.395968.39
Root analytic conductor: 8.789508.78950
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 22500, ( :9/2,9/2), 1)(4,\ 22500,\ (\ :9/2, 9/2),\ 1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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)
bad2C1C_1 (1+p4T)2 ( 1 + p^{4} T )^{2}
3C1C_1 (1+p4T)2 ( 1 + p^{4} T )^{2}
5 1 1
good7D4D_{4} 1+3166T−1852071pT2+3166p9T3+p18T4 1 + 3166 T - 1852071 p T^{2} + 3166 p^{9} T^{3} + p^{18} T^{4}
11D4D_{4} 1−27084T+4803103546T2−27084p9T3+p18T4 1 - 27084 T + 4803103546 T^{2} - 27084 p^{9} T^{3} + p^{18} T^{4}
13D4D_{4} 1+70858T+16308846387T2+70858p9T3+p18T4 1 + 70858 T + 16308846387 T^{2} + 70858 p^{9} T^{3} + p^{18} T^{4}
17D4D_{4} 1+144756T+214619001478T2+144756p9T3+p18T4 1 + 144756 T + 214619001478 T^{2} + 144756 p^{9} T^{3} + p^{18} T^{4}
19D4D_{4} 1−115690T+436265121183T2−115690p9T3+p18T4 1 - 115690 T + 436265121183 T^{2} - 115690 p^{9} T^{3} + p^{18} T^{4}
23D4D_{4} 1+799668T+2741824392082T2+799668p9T3+p18T4 1 + 799668 T + 2741824392082 T^{2} + 799668 p^{9} T^{3} + p^{18} T^{4}
29D4D_{4} 1+208140T+8938526934238T2+208140p9T3+p18T4 1 + 208140 T + 8938526934238 T^{2} + 208140 p^{9} T^{3} + p^{18} T^{4}
31D4D_{4} 1−6360094T+57405852057951T2−6360094p9T3+p18T4 1 - 6360094 T + 57405852057951 T^{2} - 6360094 p^{9} T^{3} + p^{18} T^{4}
37D4D_{4} 1+1876036T+24941642932878T2+1876036p9T3+p18T4 1 + 1876036 T + 24941642932878 T^{2} + 1876036 p^{9} T^{3} + p^{18} T^{4}
41D4D_{4} 1−38125824T+904099447138066T2−38125824p9T3+p18T4 1 - 38125824 T + 904099447138066 T^{2} - 38125824 p^{9} T^{3} + p^{18} T^{4}
43D4D_{4} 1−3776162T+682777161430647T2−3776162p9T3+p18T4 1 - 3776162 T + 682777161430647 T^{2} - 3776162 p^{9} T^{3} + p^{18} T^{4}
47D4D_{4} 1−11199924T+1294585861381378T2−11199924p9T3+p18T4 1 - 11199924 T + 1294585861381378 T^{2} - 11199924 p^{9} T^{3} + p^{18} T^{4}
53D4D_{4} 1−38780952T+6907493154005242T2−38780952p9T3+p18T4 1 - 38780952 T + 6907493154005242 T^{2} - 38780952 p^{9} T^{3} + p^{18} T^{4}
59D4D_{4} 1−286804020T+37407547383589978T2−286804020p9T3+p18T4 1 - 286804020 T + 37407547383589978 T^{2} - 286804020 p^{9} T^{3} + p^{18} T^{4}
61D4D_{4} 1+255752666T+35405174494625571T2+255752666p9T3+p18T4 1 + 255752666 T + 35405174494625571 T^{2} + 255752666 p^{9} T^{3} + p^{18} T^{4}
67D4D_{4} 1+208870006T+48155607060534303T2+208870006p9T3+p18T4 1 + 208870006 T + 48155607060534303 T^{2} + 208870006 p^{9} T^{3} + p^{18} T^{4}
71D4D_{4} 1+277057536T+105156674810622286T2+277057536p9T3+p18T4 1 + 277057536 T + 105156674810622286 T^{2} + 277057536 p^{9} T^{3} + p^{18} T^{4}
73D4D_{4} 1−60155132T+67672504711861782T2−60155132p9T3+p18T4 1 - 60155132 T + 67672504711861782 T^{2} - 60155132 p^{9} T^{3} + p^{18} T^{4}
79D4D_{4} 1−127287760T+230560963243036638T2−127287760p9T3+p18T4 1 - 127287760 T + 230560963243036638 T^{2} - 127287760 p^{9} T^{3} + p^{18} T^{4}
83D4D_{4} 1+596984628T+330798104814677002T2+596984628p9T3+p18T4 1 + 596984628 T + 330798104814677002 T^{2} + 596984628 p^{9} T^{3} + p^{18} T^{4}
89D4D_{4} 1−940116480T+562146108317908018T2−940116480p9T3+p18T4 1 - 940116480 T + 562146108317908018 T^{2} - 940116480 p^{9} T^{3} + p^{18} T^{4}
97D4D_{4} 1+1317925426T+906740973602065203T2+1317925426p9T3+p18T4 1 + 1317925426 T + 906740973602065203 T^{2} + 1317925426 p^{9} T^{3} + p^{18} 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

−10.72084476947275836962078064388, −10.66666530653370478428432025170, −9.979087594405284503781934571681, −9.598444534577307483119370754307, −9.127350034207976838821958991492, −8.597991869028760443529777816626, −7.76762530754042668138085773648, −7.48267316450309454955014130451, −6.68483232221556197914502070034, −6.54462855124113036125561547229, −5.78305907312235730302001337344, −5.37718389589092065676110937388, −4.34152289607575701726245817652, −3.95371776279156984197884995413, −2.73918520692992314118570804859, −2.39903445905845334191450592233, −1.28761801814845106503651301109, −1.04345280845314758557678611129, 0, 0, 1.04345280845314758557678611129, 1.28761801814845106503651301109, 2.39903445905845334191450592233, 2.73918520692992314118570804859, 3.95371776279156984197884995413, 4.34152289607575701726245817652, 5.37718389589092065676110937388, 5.78305907312235730302001337344, 6.54462855124113036125561547229, 6.68483232221556197914502070034, 7.48267316450309454955014130451, 7.76762530754042668138085773648, 8.597991869028760443529777816626, 9.127350034207976838821958991492, 9.598444534577307483119370754307, 9.979087594405284503781934571681, 10.66666530653370478428432025170, 10.72084476947275836962078064388

Graph of the ZZ-function along the critical line