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Properties

Label 8-637e4-1.1-c1e4-0-29
Degree 88
Conductor 164648481361164648481361
Sign 11
Analytic cond. 669.369669.369
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 3·4-s − 3·5-s + 6·6-s + 9-s + 9·10-s − 6·12-s − 14·13-s + 6·15-s − 2·16-s + 6·17-s − 3·18-s − 9·20-s − 6·23-s − 3·25-s + 42·26-s − 4·27-s + 9·29-s − 18·30-s − 30·31-s − 6·32-s − 18·34-s + 3·36-s − 24·37-s + 28·39-s − 18·41-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s + 2.44·6-s + 1/3·9-s + 2.84·10-s − 1.73·12-s − 3.88·13-s + 1.54·15-s − 1/2·16-s + 1.45·17-s − 0.707·18-s − 2.01·20-s − 1.25·23-s − 3/5·25-s + 8.23·26-s − 0.769·27-s + 1.67·29-s − 3.28·30-s − 5.38·31-s − 1.06·32-s − 3.08·34-s + 1/2·36-s − 3.94·37-s + 4.48·39-s − 2.81·41-s + ⋯

Functional equation

Λ(s)=((78⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((78⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 78⋅1347^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 669.369669.369
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 78⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad7 1 1
13C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
good2C2C_2×\timesC22C_2^2 (1+T+pT2)2(1+T−T2+pT3+p2T4) ( 1 + T + p T^{2} )^{2}( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} )
3D4D_{4} (1+T+T2+pT3+p2T4)2 ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2}
5D4×C2D_4\times C_2 1+3T+12T2+27T3+71T4+27pT5+12p2T6+3p3T7+p4T8 1 + 3 T + 12 T^{2} + 27 T^{3} + 71 T^{4} + 27 p T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−27T2+377T4−27p2T6+p4T8 1 - 27 T^{2} + 377 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1−3T−8T2−3pT3+p2T4)2 ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−31T2+537T4−31p2T6+p4T8 1 - 31 T^{2} + 537 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+6T+2T2−72T3−201T4−72pT5+2p2T6+6p3T7+p4T8 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−9T+8T2−135T3+2139T4−135pT5+8p2T6−9p3T7+p4T8 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
31C2C_2 (1+4T+pT2)2(1+11T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}
37C2C_2 (1+T+pT2)2(1+11T+pT2)2 ( 1 + T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}
41D4×C2D_4\times C_2 1+18T+210T2+1836T3+13151T4+1836pT5+210p2T6+18p3T7+p4T8 1 + 18 T + 210 T^{2} + 1836 T^{3} + 13151 T^{4} + 1836 p T^{5} + 210 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+5T−20T2−205T3−899T4−205pT5−20p2T6+5p3T7+p4T8 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1+24T+327T2+3240T3+25040T4+3240pT5+327p2T6+24p3T7+p4T8 1 + 24 T + 327 T^{2} + 3240 T^{3} + 25040 T^{4} + 3240 p T^{5} + 327 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−6T+5T2+450T3−3756T4+450pT5+5p2T6−6p3T7+p4T8 1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
59D4×C2D_4\times C_2 1−6T+21T2−54T3−2692T4−54pT5+21p2T6−6p3T7+p4T8 1 - 6 T + 21 T^{2} - 54 T^{3} - 2692 T^{4} - 54 p T^{5} + 21 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
61D4D_{4} (1−2T−66T2−2pT3+p2T4)2 ( 1 - 2 T - 66 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
67C22C_2^2×\timesC22C_2^2 (1−4T−51T2−4pT3+p2T4)(1+4T−51T2+4pT3+p2T4) ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )
71D4×C2D_4\times C_2 1−6T+150T2−828T3+14855T4−828pT5+150p2T6−6p3T7+p4T8 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
73C22C_2^2 (1−6T+85T2−6pT3+p2T4)2 ( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
79C22C_2^2 (1−6T−43T2−6pT3+p2T4)2 ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−270T2+31667T4−270p2T6+p4T8 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8}
89D4×C2D_4\times C_2 1+33T+588T2+7425T3+75011T4+7425pT5+588p2T6+33p3T7+p4T8 1 + 33 T + 588 T^{2} + 7425 T^{3} + 75011 T^{4} + 7425 p T^{5} + 588 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−39T+812T2−11895T3+132795T4−11895pT5+812p2T6−39p3T7+p4T8 1 - 39 T + 812 T^{2} - 11895 T^{3} + 132795 T^{4} - 11895 p T^{5} + 812 p^{2} T^{6} - 39 p^{3} T^{7} + p^{4} T^{8}
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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

−8.303402142257381408112723300160, −7.82587392553078210127500271384, −7.64677250007074815033225172550, −7.25343191510540273514513274793, −7.25024355995263600117105028241, −7.18905558440256622046945218427, −6.95645538376390165431511026074, −6.80135741074868912472326485978, −6.17124881041506771497473506334, −6.04682563206607060888512777965, −5.45459392373462712278001370540, −5.40875844360904347246112804037, −5.21519282429262463344276603999, −5.00027443910550228092071050657, −4.98262095179010938913852051827, −4.62374025012526072344057157228, −3.83316924665352311824692119410, −3.72838439381942006519464339291, −3.63688282479232588270682973216, −3.43820993912466889096361815679, −3.00238638961305347992632021906, −2.19183216262703043005804863449, −1.96800491764636992644598015215, −1.90064063071080855486757651577, −1.38228298275911195473528000535, 0, 0, 0, 0, 1.38228298275911195473528000535, 1.90064063071080855486757651577, 1.96800491764636992644598015215, 2.19183216262703043005804863449, 3.00238638961305347992632021906, 3.43820993912466889096361815679, 3.63688282479232588270682973216, 3.72838439381942006519464339291, 3.83316924665352311824692119410, 4.62374025012526072344057157228, 4.98262095179010938913852051827, 5.00027443910550228092071050657, 5.21519282429262463344276603999, 5.40875844360904347246112804037, 5.45459392373462712278001370540, 6.04682563206607060888512777965, 6.17124881041506771497473506334, 6.80135741074868912472326485978, 6.95645538376390165431511026074, 7.18905558440256622046945218427, 7.25024355995263600117105028241, 7.25343191510540273514513274793, 7.64677250007074815033225172550, 7.82587392553078210127500271384, 8.303402142257381408112723300160

Graph of the ZZ-function along the critical line