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Properties

Label 8-162e4-1.1-c6e4-0-1
Degree 88
Conductor 688747536688747536
Sign 11
Analytic cond. 1.92921×1061.92921\times 10^{6}
Root an. cond. 6.104816.10481
Motivic weight 66
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·4-s − 4·7-s + 3.83e3·13-s + 3.07e3·16-s + 1.02e4·19-s + 1.84e4·25-s + 256·28-s − 4.00e4·31-s − 2.02e4·37-s + 1.84e5·43-s − 4.70e5·49-s − 2.45e5·52-s + 6.09e5·61-s − 1.31e5·64-s − 2.00e6·67-s + 5.25e5·73-s − 6.55e5·76-s − 8.48e5·79-s − 1.53e4·91-s + 3.62e6·97-s − 1.17e6·100-s + 5.27e6·103-s − 1.24e6·109-s − 1.22e4·112-s + 1.95e6·121-s + 2.56e6·124-s + 127-s + ⋯
L(s)  = 1  − 4-s − 0.0116·7-s + 1.74·13-s + 3/4·16-s + 1.49·19-s + 1.17·25-s + 0.0116·28-s − 1.34·31-s − 0.398·37-s + 2.32·43-s − 3.99·49-s − 1.74·52-s + 2.68·61-s − 1/2·64-s − 6.67·67-s + 1.35·73-s − 1.49·76-s − 1.72·79-s − 0.0203·91-s + 3.96·97-s − 1.17·100-s + 4.82·103-s − 0.959·109-s − 0.00874·112-s + 1.10·121-s + 1.34·124-s − 0.0174·133-s + ⋯

Functional equation

Λ(s)=((24⋅316)s/2 ΓC(s)4 L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}
Λ(s)=((24⋅316)s/2 ΓC(s+3)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅3162^{4} \cdot 3^{16}
Sign: 11
Analytic conductor: 1.92921×1061.92921\times 10^{6}
Root analytic conductor: 6.104816.10481
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅316, ( :3,3,3,3), 1)(8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 3, 3, 3, 3 ),\ 1 )

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 4.3891411124.389141112
L(12)L(\frac12) ≈\approx 4.3891411124.389141112
L(4)L(4) 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)
bad2C2C_2 (1+p5T2)2 ( 1 + p^{5} T^{2} )^{2}
3 1 1
good5D4×C2D_4\times C_2 1−736p2T2+181599p4T4−736p14T6+p24T8 1 - 736 p^{2} T^{2} + 181599 p^{4} T^{4} - 736 p^{14} T^{6} + p^{24} T^{8}
7D4D_{4} (1+2T+235272T2+2p6T3+p12T4)2 ( 1 + 2 T + 235272 T^{2} + 2 p^{6} T^{3} + p^{12} T^{4} )^{2}
11D4×C2D_4\times C_2 1−1952680T2+729391264242T4−1952680p12T6+p24T8 1 - 1952680 T^{2} + 729391264242 T^{4} - 1952680 p^{12} T^{6} + p^{24} T^{8}
13D4D_{4} (1−1918T+7035111T2−1918p6T3+p12T4)2 ( 1 - 1918 T + 7035111 T^{2} - 1918 p^{6} T^{3} + p^{12} T^{4} )^{2}
17D4×C2D_4\times C_2 1−81233464T2+2800570401456999T4−81233464p12T6+p24T8 1 - 81233464 T^{2} + 2800570401456999 T^{4} - 81233464 p^{12} T^{6} + p^{24} T^{8}
19D4D_{4} (1−5122T+92571840T2−5122p6T3+p12T4)2 ( 1 - 5122 T + 92571840 T^{2} - 5122 p^{6} T^{3} + p^{12} T^{4} )^{2}
23D4×C2D_4\times C_2 1−306194656T2+48390197876780226T4−306194656p12T6+p24T8 1 - 306194656 T^{2} + 48390197876780226 T^{4} - 306194656 p^{12} T^{6} + p^{24} T^{8}
29D4×C2D_4\times C_2 1+218801912T2+713450102444654343T4+218801912p12T6+p24T8 1 + 218801912 T^{2} + 713450102444654343 T^{4} + 218801912 p^{12} T^{6} + p^{24} T^{8}
31D4D_{4} (1+20048T+1659858486T2+20048p6T3+p12T4)2 ( 1 + 20048 T + 1659858486 T^{2} + 20048 p^{6} T^{3} + p^{12} T^{4} )^{2}
37D4D_{4} (1+10100T+5013752475T2+10100p6T3+p12T4)2 ( 1 + 10100 T + 5013752475 T^{2} + 10100 p^{6} T^{3} + p^{12} T^{4} )^{2}
41D4×C2D_4\times C_2 1−2589640888T2+18656732024545255890T4−2589640888p12T6+p24T8 1 - 2589640888 T^{2} + 18656732024545255890 T^{4} - 2589640888 p^{12} T^{6} + p^{24} T^{8}
43D4D_{4} (1−92470T+8867120496T2−92470p6T3+p12T4)2 ( 1 - 92470 T + 8867120496 T^{2} - 92470 p^{6} T^{3} + p^{12} T^{4} )^{2}
47D4×C2D_4\times C_2 1−27858666964T2+191878190860258806p2T4−27858666964p12T6+p24T8 1 - 27858666964 T^{2} + 191878190860258806 p^{2} T^{4} - 27858666964 p^{12} T^{6} + p^{24} T^{8}
53D4×C2D_4\times C_2 1−51925621720T2+ 1 - 51925621720 T^{2} + 16 ⁣⋯ ⁣9416\!\cdots\!94T4−51925621720p12T6+p24T8 T^{4} - 51925621720 p^{12} T^{6} + p^{24} T^{8}
59D4×C2D_4\times C_2 1−144480622612T2+ 1 - 144480622612 T^{2} + 86 ⁣⋯ ⁣2686\!\cdots\!26T4−144480622612p12T6+p24T8 T^{4} - 144480622612 p^{12} T^{6} + p^{24} T^{8}
61D4D_{4} (1−304528T+97865805831T2−304528p6T3+p12T4)2 ( 1 - 304528 T + 97865805831 T^{2} - 304528 p^{6} T^{3} + p^{12} T^{4} )^{2}
67D4D_{4} (1+1004486T+432404360664T2+1004486p6T3+p12T4)2 ( 1 + 1004486 T + 432404360664 T^{2} + 1004486 p^{6} T^{3} + p^{12} T^{4} )^{2}
71D4×C2D_4\times C_2 1−94765336960T2+ 1 - 94765336960 T^{2} + 38 ⁣⋯ ⁣7438\!\cdots\!74T4−94765336960p12T6+p24T8 T^{4} - 94765336960 p^{12} T^{6} + p^{24} T^{8}
73D4D_{4} (1−262912T+52515591p2T2−262912p6T3+p12T4)2 ( 1 - 262912 T + 52515591 p^{2} T^{2} - 262912 p^{6} T^{3} + p^{12} T^{4} )^{2}
79D4D_{4} (1+424358T+63430257600T2+424358p6T3+p12T4)2 ( 1 + 424358 T + 63430257600 T^{2} + 424358 p^{6} T^{3} + p^{12} T^{4} )^{2}
83D4×C2D_4\times C_2 1−413441187988T2+ 1 - 413441187988 T^{2} + 57 ⁣⋯ ⁣8657\!\cdots\!86T4−413441187988p12T6+p24T8 T^{4} - 413441187988 p^{12} T^{6} + p^{24} T^{8}
89D4×C2D_4\times C_2 1−332250007216T2+ 1 - 332250007216 T^{2} + 48 ⁣⋯ ⁣5948\!\cdots\!59T4−332250007216p12T6+p24T8 T^{4} - 332250007216 p^{12} T^{6} + p^{24} T^{8}
97D4D_{4} (1−1810864T+2248345168182T2−1810864p6T3+p12T4)2 ( 1 - 1810864 T + 2248345168182 T^{2} - 1810864 p^{6} T^{3} + p^{12} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.470500397925486215641505829311, −7.965692940750314557097277917638, −7.69911966681077427095742372652, −7.40765341690456207880144272247, −7.38360132305516723843919135754, −6.83792085026262081554111260614, −6.58877983235400973149546385791, −6.04789163722634940481002238358, −6.01337324632194373119299402676, −5.67136035591161315748859067512, −5.51357133450215452671542029177, −4.87354585465978095580975210731, −4.70088089819609813924858913189, −4.55643415361811843409152634271, −4.14925118757044234608576974113, −3.55102942194812949667365946150, −3.41748306930860739481483671933, −3.15585639334753910017239072969, −2.97299123776464371834841582617, −2.15065574576836204074426493832, −1.75135500212716955847425025014, −1.44217208066409831776983601739, −1.02765213242165244982656942075, −0.60825224092598202943167468638, −0.38030557152875619794465872017, 0.38030557152875619794465872017, 0.60825224092598202943167468638, 1.02765213242165244982656942075, 1.44217208066409831776983601739, 1.75135500212716955847425025014, 2.15065574576836204074426493832, 2.97299123776464371834841582617, 3.15585639334753910017239072969, 3.41748306930860739481483671933, 3.55102942194812949667365946150, 4.14925118757044234608576974113, 4.55643415361811843409152634271, 4.70088089819609813924858913189, 4.87354585465978095580975210731, 5.51357133450215452671542029177, 5.67136035591161315748859067512, 6.01337324632194373119299402676, 6.04789163722634940481002238358, 6.58877983235400973149546385791, 6.83792085026262081554111260614, 7.38360132305516723843919135754, 7.40765341690456207880144272247, 7.69911966681077427095742372652, 7.965692940750314557097277917638, 8.470500397925486215641505829311

Graph of the ZZ-function along the critical line