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Properties

Label 8-975e4-1.1-c1e4-0-18
Degree 88
Conductor 903687890625903687890625
Sign 11
Analytic cond. 3673.893673.89
Root an. cond. 2.790232.79023
Motivic weight 11
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  + 2·2-s + 2·4-s + 2·7-s − 4·8-s + 9-s − 12·11-s − 14·13-s + 4·14-s − 12·16-s − 6·17-s + 2·18-s + 12·19-s − 24·22-s + 12·23-s − 28·26-s + 4·28-s − 2·29-s − 16·32-s − 12·34-s + 2·36-s + 8·37-s + 24·38-s − 6·41-s − 6·43-s − 24·44-s + 24·46-s − 20·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.755·7-s − 1.41·8-s + 1/3·9-s − 3.61·11-s − 3.88·13-s + 1.06·14-s − 3·16-s − 1.45·17-s + 0.471·18-s + 2.75·19-s − 5.11·22-s + 2.50·23-s − 5.49·26-s + 0.755·28-s − 0.371·29-s − 2.82·32-s − 2.05·34-s + 1/3·36-s + 1.31·37-s + 3.89·38-s − 0.937·41-s − 0.914·43-s − 3.61·44-s + 3.53·46-s − 2.91·47-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅58⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{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: 34⋅58⋅1343^{4} \cdot 5^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 3673.893673.89
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.4730488441.473048844
L(12)L(\frac12) ≈\approx 1.4730488441.473048844
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)
bad3C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5 1 1
13C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
good2C2C_2×\timesC22C_2^2 (1−pT+pT2)2(1+pT+pT2+p2T3+p2T4) ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )
7D4×C2D_4\times C_2 1−2T+T2+22T3−68T4+22pT5+p2T6−2p3T7+p4T8 1 - 2 T + T^{2} + 22 T^{3} - 68 T^{4} + 22 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
11C22C_2^2 (1+6T+23T2+6pT3+p2T4)2 ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
17D4×C2D_4\times C_2 1+6T+24T2+72T3+59T4+72pT5+24p2T6+6p3T7+p4T8 1 + 6 T + 24 T^{2} + 72 T^{3} + 59 T^{4} + 72 p T^{5} + 24 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−12T+94T2−552T3+2667T4−552pT5+94p2T6−12p3T7+p4T8 1 - 12 T + 94 T^{2} - 552 T^{3} + 2667 T^{4} - 552 p T^{5} + 94 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−12T+90T2−504T3+2339T4−504pT5+90p2T6−12p3T7+p4T8 1 - 12 T + 90 T^{2} - 504 T^{3} + 2339 T^{4} - 504 p T^{5} + 90 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+2T−52T2−4T3+2179T4−4pT5−52p2T6+2p3T7+p4T8 1 + 2 T - 52 T^{2} - 4 T^{3} + 2179 T^{4} - 4 p T^{5} - 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−2pT2+2451T4−2p3T6+p4T8 1 - 2 p T^{2} + 2451 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8}
37C22C_2^2 (1−4T−21T2−4pT3+p2T4)2 ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
41D4×C2D_4\times C_2 1+6T+48T2+216T3+107T4+216pT5+48p2T6+6p3T7+p4T8 1 + 6 T + 48 T^{2} + 216 T^{3} + 107 T^{4} + 216 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+6T+97T2+510T3+5892T4+510pT5+97p2T6+6p3T7+p4T8 1 + 6 T + 97 T^{2} + 510 T^{3} + 5892 T^{4} + 510 p T^{5} + 97 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
47D4D_{4} (1+10T+92T2+10pT3+p2T4)2 ( 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−58T2+p2T4)2 ( 1 - 58 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−6T+52T2−240T3−1173T4−240pT5+52p2T6−6p3T7+p4T8 1 - 6 T + 52 T^{2} - 240 T^{3} - 1173 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+2T−107T2−22T3+8356T4−22pT5−107p2T6+2p3T7+p4T8 1 + 2 T - 107 T^{2} - 22 T^{3} + 8356 T^{4} - 22 p T^{5} - 107 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1+18T+121T2+1242T3+15012T4+1242pT5+121p2T6+18p3T7+p4T8 1 + 18 T + 121 T^{2} + 1242 T^{3} + 15012 T^{4} + 1242 p T^{5} + 121 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1+6T+36T2+144T3−3613T4+144pT5+36p2T6+6p3T7+p4T8 1 + 6 T + 36 T^{2} + 144 T^{3} - 3613 T^{4} + 144 p T^{5} + 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
73D4D_{4} (1+10T+63T2+10pT3+p2T4)2 ( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
79D4D_{4} (1−10T+135T2−10pT3+p2T4)2 ( 1 - 10 T + 135 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
83D4D_{4} (1−12T+190T2−12pT3+p2T4)2 ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1+6T+184T2+1032T3+22731T4+1032pT5+184p2T6+6p3T7+p4T8 1 + 6 T + 184 T^{2} + 1032 T^{3} + 22731 T^{4} + 1032 p T^{5} + 184 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−26T+325T2−4082T3+48220T4−4082pT5+325p2T6−26p3T7+p4T8 1 - 26 T + 325 T^{2} - 4082 T^{3} + 48220 T^{4} - 4082 p T^{5} + 325 p^{2} T^{6} - 26 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

−7.22577491817874276141009544835, −7.02172669617890531789811456920, −6.81778308920199025808024835077, −6.38492724332742250949716171348, −6.14310973732141578409462640590, −6.05727543774034306811299306411, −5.48822805131380004413032203107, −5.38576324195486280145212708975, −5.23084654268449229108372876199, −4.96778285728543956968355378161, −4.90721015185146233320028994043, −4.87711295760923943476710875680, −4.53785950985929038582079028864, −4.49240510380355209543938562531, −3.71811270813407830479229224868, −3.37776824214811318130499538891, −3.14936067752494433759763420025, −2.92109712901595911801998121848, −2.82968354386930019317707583891, −2.69303738149419749946921510041, −2.17225448219348128793023997758, −2.11469934058114748316153610013, −1.65485661416794654420197058996, −0.59413683218181612085849560278, −0.31248525208518421653984280236, 0.31248525208518421653984280236, 0.59413683218181612085849560278, 1.65485661416794654420197058996, 2.11469934058114748316153610013, 2.17225448219348128793023997758, 2.69303738149419749946921510041, 2.82968354386930019317707583891, 2.92109712901595911801998121848, 3.14936067752494433759763420025, 3.37776824214811318130499538891, 3.71811270813407830479229224868, 4.49240510380355209543938562531, 4.53785950985929038582079028864, 4.87711295760923943476710875680, 4.90721015185146233320028994043, 4.96778285728543956968355378161, 5.23084654268449229108372876199, 5.38576324195486280145212708975, 5.48822805131380004413032203107, 6.05727543774034306811299306411, 6.14310973732141578409462640590, 6.38492724332742250949716171348, 6.81778308920199025808024835077, 7.02172669617890531789811456920, 7.22577491817874276141009544835

Graph of the ZZ-function along the critical line