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Properties

Label 4-396e2-1.1-c3e2-0-2
Degree 44
Conductor 156816156816
Sign 11
Analytic cond. 545.911545.911
Root an. cond. 4.833714.83371
Motivic weight 33
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  + 12·5-s − 24·7-s − 22·11-s − 8·13-s − 28·17-s − 68·19-s − 268·23-s − 18·25-s − 260·29-s + 112·31-s − 288·35-s + 316·37-s − 124·41-s − 148·43-s − 572·47-s − 130·49-s − 76·53-s − 264·55-s − 864·59-s − 136·61-s − 96·65-s − 512·67-s − 724·71-s + 972·73-s + 528·77-s − 528·79-s − 2.11e3·83-s + ⋯
L(s)  = 1  + 1.07·5-s − 1.29·7-s − 0.603·11-s − 0.170·13-s − 0.399·17-s − 0.821·19-s − 2.42·23-s − 0.143·25-s − 1.66·29-s + 0.648·31-s − 1.39·35-s + 1.40·37-s − 0.472·41-s − 0.524·43-s − 1.77·47-s − 0.379·49-s − 0.196·53-s − 0.647·55-s − 1.90·59-s − 0.285·61-s − 0.183·65-s − 0.933·67-s − 1.21·71-s + 1.55·73-s + 0.781·77-s − 0.751·79-s − 2.79·83-s + ⋯

Functional equation

Λ(s)=(156816s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(156816s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 156816156816    =    24⋅34⋅1122^{4} \cdot 3^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 545.911545.911
Root analytic conductor: 4.833714.83371
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 156816, ( :3/2,3/2), 1)(4,\ 156816,\ (\ :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
3 1 1
11C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1−12T+162T2−12p3T3+p6T4 1 - 12 T + 162 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+24T+706T2+24p3T3+p6T4 1 + 24 T + 706 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+8T+1310T2+8p3T3+p6T4 1 + 8 T + 1310 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+28T+5558T2+28p3T3+p6T4 1 + 28 T + 5558 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+68T+14378T2+68p3T3+p6T4 1 + 68 T + 14378 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+268T+36214T2+268p3T3+p6T4 1 + 268 T + 36214 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+260T+63694T2+260p3T3+p6T4 1 + 260 T + 63694 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−112T+38414T2−112p3T3+p6T4 1 - 112 T + 38414 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−316T+66254T2−316p3T3+p6T4 1 - 316 T + 66254 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+124T+133750T2+124p3T3+p6T4 1 + 124 T + 133750 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4}
43C2C_2 (1+74T+p3T2)2 ( 1 + 74 T + p^{3} T^{2} )^{2}
47D4D_{4} 1+572T+274438T2+572p3T3+p6T4 1 + 572 T + 274438 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+76T+48098T2+76p3T3+p6T4 1 + 76 T + 48098 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+864T+592918T2+864p3T3+p6T4 1 + 864 T + 592918 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+136T+136062T2+136p3T3+p6T4 1 + 136 T + 136062 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+512T+468662T2+512p3T3+p6T4 1 + 512 T + 468662 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+724T+694966T2+724p3T3+p6T4 1 + 724 T + 694966 T^{2} + 724 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−972T+996374T2−972p3T3+p6T4 1 - 972 T + 996374 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+528T+1052674T2+528p3T3+p6T4 1 + 528 T + 1052674 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+2112T+2198694T2+2112p3T3+p6T4 1 + 2112 T + 2198694 T^{2} + 2112 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+288T−124782T2+288p3T3+p6T4 1 + 288 T - 124782 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−500T+1245030T2−500p3T3+p6T4 1 - 500 T + 1245030 T^{2} - 500 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

−10.40529560901769179360940352977, −10.05783308804332371999961816570, −9.691943699057387933092912451342, −9.602780152248101710406336583547, −8.904757943762688612686988469143, −8.305205710953796645705507284429, −7.87811740800547413327798666806, −7.41064726322153901475281195952, −6.50910591695764302787481839212, −6.42913890955334638661628131567, −5.76781803290502380869367139818, −5.66994616577643587810566789130, −4.56896407020150818853070963648, −4.28351314502518495303545676599, −3.36512440568639058159585179703, −2.91448017618907003083281129995, −2.01237317689166792041705731153, −1.74515037331068871195572042549, 0, 0, 1.74515037331068871195572042549, 2.01237317689166792041705731153, 2.91448017618907003083281129995, 3.36512440568639058159585179703, 4.28351314502518495303545676599, 4.56896407020150818853070963648, 5.66994616577643587810566789130, 5.76781803290502380869367139818, 6.42913890955334638661628131567, 6.50910591695764302787481839212, 7.41064726322153901475281195952, 7.87811740800547413327798666806, 8.305205710953796645705507284429, 8.904757943762688612686988469143, 9.602780152248101710406336583547, 9.691943699057387933092912451342, 10.05783308804332371999961816570, 10.40529560901769179360940352977

Graph of the ZZ-function along the critical line