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Properties

Label 4-24e4-1.1-c7e2-0-15
Degree 44
Conductor 331776331776
Sign 11
Analytic cond. 32376.132376.1
Root an. cond. 13.413913.4139
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 520·7-s − 1.37e4·13-s + 6.63e4·19-s − 6.40e4·25-s − 3.01e3·31-s + 7.61e5·37-s + 1.52e4·43-s − 1.44e6·49-s + 1.97e6·61-s + 7.71e6·67-s − 4.00e6·73-s − 5.39e6·79-s + 7.16e6·91-s − 2.59e7·97-s − 1.01e7·103-s − 1.31e7·109-s − 2.11e6·121-s + 127-s + 131-s − 3.45e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.573·7-s − 1.73·13-s + 2.21·19-s − 0.820·25-s − 0.0181·31-s + 2.47·37-s + 0.0293·43-s − 1.75·49-s + 1.11·61-s + 3.13·67-s − 1.20·73-s − 1.23·79-s + 0.996·91-s − 2.88·97-s − 0.914·103-s − 0.975·109-s − 0.108·121-s − 1.27·133-s + 0.269·169-s + ⋯

Functional equation

Λ(s)=(331776s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(331776s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 331776331776    =    212⋅342^{12} \cdot 3^{4}
Sign: 11
Analytic conductor: 32376.132376.1
Root analytic conductor: 13.413913.4139
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 331776, ( :7/2,7/2), 1)(4,\ 331776,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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
3 1 1
good5C22C_2^2 1+12818pT2+p14T4 1 + 12818 p T^{2} + p^{14} T^{4}
7C2C_2 (1+260T+p7T2)2 ( 1 + 260 T + p^{7} T^{2} )^{2}
11C22C_2^2 1+2110342T2+p14T4 1 + 2110342 T^{2} + p^{14} T^{4}
13C2C_2 (1+530pT+p7T2)2 ( 1 + 530 p T + p^{7} T^{2} )^{2}
17C22C_2^2 1+259975906T2+p14T4 1 + 259975906 T^{2} + p^{14} T^{4}
19C2C_2 (1−33176T+p7T2)2 ( 1 - 33176 T + p^{7} T^{2} )^{2}
23C22C_2^2 1+5812848334T2+p14T4 1 + 5812848334 T^{2} + p^{14} T^{4}
29C22C_2^2 1+15420328618T2+p14T4 1 + 15420328618 T^{2} + p^{14} T^{4}
31C2C_2 (1+1508T+p7T2)2 ( 1 + 1508 T + p^{7} T^{2} )^{2}
37C2C_2 (1−380770T+p7T2)2 ( 1 - 380770 T + p^{7} T^{2} )^{2}
41C22C_2^2 1+381757891762T2+p14T4 1 + 381757891762 T^{2} + p^{14} T^{4}
43C2C_2 (1−7640T+p7T2)2 ( 1 - 7640 T + p^{7} T^{2} )^{2}
47C22C_2^2 1+693036689566T2+p14T4 1 + 693036689566 T^{2} + p^{14} T^{4}
53C22C_2^2 1+1288434979834T2+p14T4 1 + 1288434979834 T^{2} + p^{14} T^{4}
59C22C_2^2 1−2355536454362T2+p14T4 1 - 2355536454362 T^{2} + p^{14} T^{4}
61C2C_2 (1−988858T+p7T2)2 ( 1 - 988858 T + p^{7} T^{2} )^{2}
67C2C_2 (1−3857360T+p7T2)2 ( 1 - 3857360 T + p^{7} T^{2} )^{2}
71C22C_2^2 1+332728892782T2+p14T4 1 + 332728892782 T^{2} + p^{14} T^{4}
73C2C_2 (1+2004730T+p7T2)2 ( 1 + 2004730 T + p^{7} T^{2} )^{2}
79C2C_2 (1+2699684T+p7T2)2 ( 1 + 2699684 T + p^{7} T^{2} )^{2}
83C22C_2^2 1+46919519671414T2+p14T4 1 + 46919519671414 T^{2} + p^{14} T^{4}
89C22C_2^2 1+28535629791058T2+p14T4 1 + 28535629791058 T^{2} + p^{14} T^{4}
97C2C_2 (1+12957490T+p7T2)2 ( 1 + 12957490 T + p^{7} T^{2} )^{2}
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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

−9.609245929252774116092130248649, −9.221859956455362371633823458944, −8.260855370354819093488253266429, −8.111452285143942220215996287628, −7.50850658010594023630425439633, −7.25549151194211566254629968668, −6.72852225184488612790515932904, −6.28102814052563532841224822124, −5.49910126988007551411248075087, −5.42879566852901528794226898365, −4.78073455698241697806145303348, −4.27813734619205883733352703719, −3.65953067931361611130374058525, −3.16182731428532129268243957028, −2.45376117781314261962851373677, −2.43617660095618298606537856012, −1.27444247989618622872836836205, −1.03706092418535297370139287557, 0, 0, 1.03706092418535297370139287557, 1.27444247989618622872836836205, 2.43617660095618298606537856012, 2.45376117781314261962851373677, 3.16182731428532129268243957028, 3.65953067931361611130374058525, 4.27813734619205883733352703719, 4.78073455698241697806145303348, 5.42879566852901528794226898365, 5.49910126988007551411248075087, 6.28102814052563532841224822124, 6.72852225184488612790515932904, 7.25549151194211566254629968668, 7.50850658010594023630425439633, 8.111452285143942220215996287628, 8.260855370354819093488253266429, 9.221859956455362371633823458944, 9.609245929252774116092130248649

Graph of the ZZ-function along the critical line