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Properties

Label 4-312e2-1.1-c3e2-0-4
Degree 44
Conductor 9734497344
Sign 11
Analytic cond. 338.876338.876
Root an. cond. 4.290524.29052
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  + 6·3-s − 18·5-s − 10·7-s + 27·9-s + 4·11-s + 26·13-s − 108·15-s − 52·17-s − 142·19-s − 60·21-s − 280·23-s + 10·25-s + 108·27-s − 424·29-s − 178·31-s + 24·33-s + 180·35-s + 136·37-s + 156·39-s − 226·41-s − 72·43-s − 486·45-s + 176·47-s − 186·49-s − 312·51-s − 788·53-s − 72·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.60·5-s − 0.539·7-s + 9-s + 0.109·11-s + 0.554·13-s − 1.85·15-s − 0.741·17-s − 1.71·19-s − 0.623·21-s − 2.53·23-s + 2/25·25-s + 0.769·27-s − 2.71·29-s − 1.03·31-s + 0.126·33-s + 0.869·35-s + 0.604·37-s + 0.640·39-s − 0.860·41-s − 0.255·43-s − 1.60·45-s + 0.546·47-s − 0.542·49-s − 0.856·51-s − 2.04·53-s − 0.176·55-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 338.876338.876
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 97344, ( :3/2,3/2), 1)(4,\ 97344,\ (\ :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+18T+314T2+18p3T3+p6T4 1 + 18 T + 314 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+10T+286T2+10p3T3+p6T4 1 + 10 T + 286 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−4T−666T2−4p3T3+p6T4 1 - 4 T - 666 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+52T+8054T2+52p3T3+p6T4 1 + 52 T + 8054 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+142T+16702T2+142p3T3+p6T4 1 + 142 T + 16702 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+280T+41486T2+280p3T3+p6T4 1 + 280 T + 41486 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+424T+93110T2+424p3T3+p6T4 1 + 424 T + 93110 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+178T+48990T2+178p3T3+p6T4 1 + 178 T + 48990 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−136T+56358T2−136p3T3+p6T4 1 - 136 T + 56358 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+226T+144474T2+226p3T3+p6T4 1 + 226 T + 144474 T^{2} + 226 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+72T+28662T2+72p3T3+p6T4 1 + 72 T + 28662 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−176T−29410T2−176p3T3+p6T4 1 - 176 T - 29410 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+788T+451902T2+788p3T3+p6T4 1 + 788 T + 451902 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−728T+542166T2−728p3T3+p6T4 1 - 728 T + 542166 T^{2} - 728 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+736T+564838T2+736p3T3+p6T4 1 + 736 T + 564838 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−1054T+684622T2−1054p3T3+p6T4 1 - 1054 T + 684622 T^{2} - 1054 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+660T+721294T2+660p3T3+p6T4 1 + 660 T + 721294 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−24T−59650T2−24p3T3+p6T4 1 - 24 T - 59650 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−40T−308514T2−40p3T3+p6T4 1 - 40 T - 308514 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−1344T+1516550T2−1344p3T3+p6T4 1 - 1344 T + 1516550 T^{2} - 1344 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−314T+619250T2−314p3T3+p6T4 1 - 314 T + 619250 T^{2} - 314 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+264T+1841070T2+264p3T3+p6T4 1 + 264 T + 1841070 T^{2} + 264 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

−10.83394782654989645206140862298, −10.79479718180286426519951884760, −9.930619319756160097953807829626, −9.472805026810281246778095833868, −9.147686717995460193798278020972, −8.486485898348483876344336995153, −7.985894070032192047860297427511, −7.965347053620023151782253387617, −7.32760487574813187890454752491, −6.72377644769637053377010965107, −6.17343318526163125708986232958, −5.63734182546104978125567275658, −4.46759038480210206541089976156, −4.14240725570613377864073640081, −3.60831962071639935710322882913, −3.45709916490424119660510429234, −2.00893613230940240624886554959, −2.00838632588726881683853527300, 0, 0, 2.00838632588726881683853527300, 2.00893613230940240624886554959, 3.45709916490424119660510429234, 3.60831962071639935710322882913, 4.14240725570613377864073640081, 4.46759038480210206541089976156, 5.63734182546104978125567275658, 6.17343318526163125708986232958, 6.72377644769637053377010965107, 7.32760487574813187890454752491, 7.965347053620023151782253387617, 7.985894070032192047860297427511, 8.486485898348483876344336995153, 9.147686717995460193798278020972, 9.472805026810281246778095833868, 9.930619319756160097953807829626, 10.79479718180286426519951884760, 10.83394782654989645206140862298

Graph of the ZZ-function along the critical line