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Properties

Label 4-624e2-1.1-c3e2-0-17
Degree 44
Conductor 389376389376
Sign 11
Analytic cond. 1355.501355.50
Root an. cond. 6.067716.06771
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  + 6·3-s − 4·5-s − 4·7-s + 27·9-s − 28·11-s − 26·13-s − 24·15-s − 36·17-s − 44·19-s − 24·21-s + 8·23-s − 226·25-s + 108·27-s − 204·29-s + 164·31-s − 168·33-s + 16·35-s − 668·37-s − 156·39-s − 100·41-s + 272·43-s − 108·45-s − 60·47-s − 566·49-s − 216·51-s − 708·53-s + 112·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.357·5-s − 0.215·7-s + 9-s − 0.767·11-s − 0.554·13-s − 0.413·15-s − 0.513·17-s − 0.531·19-s − 0.249·21-s + 0.0725·23-s − 1.80·25-s + 0.769·27-s − 1.30·29-s + 0.950·31-s − 0.886·33-s + 0.0772·35-s − 2.96·37-s − 0.640·39-s − 0.380·41-s + 0.964·43-s − 0.357·45-s − 0.186·47-s − 1.65·49-s − 0.593·51-s − 1.83·53-s + 0.274·55-s + ⋯

Functional equation

Λ(s)=(389376s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(389376s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 389376389376    =    28⋅32⋅1322^{8} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 1355.501355.50
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 389376, ( :3/2,3/2), 1)(4,\ 389376,\ (\ :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)
bad2 1 1
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1+4T+242T2+4p3T3+p6T4 1 + 4 T + 242 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+4T+582T2+4p3T3+p6T4 1 + 4 T + 582 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+28T+2426T2+28p3T3+p6T4 1 + 28 T + 2426 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+36T+2038T2+36p3T3+p6T4 1 + 36 T + 2038 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+44T−498T2+44p3T3+p6T4 1 + 44 T - 498 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−8T+8798T2−8p3T3+p6T4 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+204T+52270T2+204p3T3+p6T4 1 + 204 T + 52270 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−164T+66294T2−164p3T3+p6T4 1 - 164 T + 66294 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+668T+212814T2+668p3T3+p6T4 1 + 668 T + 212814 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+100T−2230T2+100p3T3+p6T4 1 + 100 T - 2230 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−272T+137142T2−272p3T3+p6T4 1 - 272 T + 137142 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+60T+203746T2+60p3T3+p6T4 1 + 60 T + 203746 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+708T+312478T2+708p3T3+p6T4 1 + 708 T + 312478 T^{2} + 708 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−180T+395626T2−180p3T3+p6T4 1 - 180 T + 395626 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+1068T+726830T2+1068p3T3+p6T4 1 + 1068 T + 726830 T^{2} + 1068 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−420T+207854T2−420p3T3+p6T4 1 - 420 T + 207854 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+436T+749474T2+436p3T3+p6T4 1 + 436 T + 749474 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+412T+163398T2+412p3T3+p6T4 1 + 412 T + 163398 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+672T+559646T2+672p3T3+p6T4 1 + 672 T + 559646 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−124T+501530T2−124p3T3+p6T4 1 - 124 T + 501530 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−140T+1088138T2−140p3T3+p6T4 1 - 140 T + 1088138 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+188T−343530T2+188p3T3+p6T4 1 + 188 T - 343530 T^{2} + 188 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

−9.789258739575076852169930477637, −9.694363709486846408786170940517, −8.954145579524072772818608161960, −8.792655390419103954979094097120, −8.148080423537310633559375186704, −7.88238360860659626711853741046, −7.35033661856113913683140544656, −7.19256751390995337792015543616, −6.23462255446169174667819699184, −6.19793003563150396067527806091, −5.12286682976149393526029596432, −4.96405277157720327117499920335, −4.18058544513800717697195864857, −3.77054670889504364859528607188, −3.17571619151732251100228736435, −2.72714156116705116800813935824, −1.88577486887734932884279074016, −1.66822073065225087140835287046, 0, 0, 1.66822073065225087140835287046, 1.88577486887734932884279074016, 2.72714156116705116800813935824, 3.17571619151732251100228736435, 3.77054670889504364859528607188, 4.18058544513800717697195864857, 4.96405277157720327117499920335, 5.12286682976149393526029596432, 6.19793003563150396067527806091, 6.23462255446169174667819699184, 7.19256751390995337792015543616, 7.35033661856113913683140544656, 7.88238360860659626711853741046, 8.148080423537310633559375186704, 8.792655390419103954979094097120, 8.954145579524072772818608161960, 9.694363709486846408786170940517, 9.789258739575076852169930477637

Graph of the ZZ-function along the critical line