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Properties

Label 4-832e2-1.1-c3e2-0-12
Degree 44
Conductor 692224692224
Sign 11
Analytic cond. 2409.782409.78
Root an. cond. 7.006397.00639
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  − 5·3-s + 3·5-s − 9·7-s + 3·9-s − 80·11-s + 26·13-s − 15·15-s + 19·17-s + 84·19-s + 45·21-s + 196·23-s − 239·25-s − 40·27-s + 44·29-s − 86·31-s + 400·33-s − 27·35-s − 209·37-s − 130·39-s − 230·41-s − 287·43-s + 9·45-s + 435·47-s − 111·49-s − 95·51-s + 118·53-s − 240·55-s + ⋯
L(s)  = 1  − 0.962·3-s + 0.268·5-s − 0.485·7-s + 1/9·9-s − 2.19·11-s + 0.554·13-s − 0.258·15-s + 0.271·17-s + 1.01·19-s + 0.467·21-s + 1.77·23-s − 1.91·25-s − 0.285·27-s + 0.281·29-s − 0.498·31-s + 2.11·33-s − 0.130·35-s − 0.928·37-s − 0.533·39-s − 0.876·41-s − 1.01·43-s + 0.0298·45-s + 1.35·47-s − 0.323·49-s − 0.260·51-s + 0.305·53-s − 0.588·55-s + ⋯

Functional equation

Λ(s)=(692224s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(692224s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 692224692224    =    212⋅1322^{12} \cdot 13^{2}
Sign: 11
Analytic conductor: 2409.782409.78
Root analytic conductor: 7.006397.00639
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 692224, ( :3/2,3/2), 1)(4,\ 692224,\ (\ :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
13C1C_1 (1−pT)2 ( 1 - p T )^{2}
good3D4D_{4} 1+5T+22T2+5p3T3+p6T4 1 + 5 T + 22 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4}
5D4D_{4} 1−3T+248T2−3p3T3+p6T4 1 - 3 T + 248 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+9T+192T2+9p3T3+p6T4 1 + 9 T + 192 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+80T+3650T2+80p3T3+p6T4 1 + 80 T + 3650 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−19T+8688T2−19p3T3+p6T4 1 - 19 T + 8688 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−84T+11130T2−84p3T3+p6T4 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−196T+33326T2−196p3T3+p6T4 1 - 196 T + 33326 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−44T+10094T2−44p3T3+p6T4 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+86T+56518T2+86p3T3+p6T4 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+209T+112120T2+209p3T3+p6T4 1 + 209 T + 112120 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+230T+149010T2+230p3T3+p6T4 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+287T+92698T2+287p3T3+p6T4 1 + 287 T + 92698 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−435T+192728T2−435p3T3+p6T4 1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−118T+297410T2−118p3T3+p6T4 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−368T+379266T2−368p3T3+p6T4 1 - 368 T + 379266 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−1058T+580378T2−1058p3T3+p6T4 1 - 1058 T + 580378 T^{2} - 1058 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+68T+373930T2+68p3T3+p6T4 1 + 68 T + 373930 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+131T+493328T2+131p3T3+p6T4 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−456T+542718T2−456p3T3+p6T4 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+1008T+1233294T2+1008p3T3+p6T4 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1958T+1961238T2+1958p3T3+p6T4 1 + 1958 T + 1961238 T^{2} + 1958 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+720T+899726T2+720p3T3+p6T4 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+928T+943870T2+928p3T3+p6T4 1 + 928 T + 943870 T^{2} + 928 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

−9.813278967619206055679723237484, −9.319499284172354490094845639702, −8.690575745351321752582709298952, −8.331002260396050721010931764949, −7.935250529032507360428747065525, −7.27826445201795444998474985548, −7.11145528861906989029965312862, −6.58508725825674101626681817323, −5.89484740307466743700789038238, −5.50580407107914811657684156438, −5.23980109933732563050668266044, −5.16052919469470520776141001360, −4.06525882184495625362183936423, −3.70945114684107574543167233522, −2.81311815052099493056604838131, −2.74366042504932949051645542020, −1.74628020280605270243765824030, −1.05238160253304729636769076051, 0, 0, 1.05238160253304729636769076051, 1.74628020280605270243765824030, 2.74366042504932949051645542020, 2.81311815052099493056604838131, 3.70945114684107574543167233522, 4.06525882184495625362183936423, 5.16052919469470520776141001360, 5.23980109933732563050668266044, 5.50580407107914811657684156438, 5.89484740307466743700789038238, 6.58508725825674101626681817323, 7.11145528861906989029965312862, 7.27826445201795444998474985548, 7.935250529032507360428747065525, 8.331002260396050721010931764949, 8.690575745351321752582709298952, 9.319499284172354490094845639702, 9.813278967619206055679723237484

Graph of the ZZ-function along the critical line