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Properties

Label 8-525e4-1.1-c1e4-0-17
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 308.848308.848
Root an. cond. 2.047472.04747
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 18·4-s + 10·7-s − 36·8-s + 9-s + 2·11-s − 60·14-s + 52·16-s + 6·17-s − 6·18-s + 2·19-s − 12·22-s + 6·23-s + 180·28-s − 4·29-s − 6·31-s − 48·32-s − 36·34-s + 18·36-s − 18·37-s − 12·38-s + 4·41-s + 36·44-s − 36·46-s + 61·49-s + 36·53-s − 360·56-s + ⋯
L(s)  = 1  − 4.24·2-s + 9·4-s + 3.77·7-s − 12.7·8-s + 1/3·9-s + 0.603·11-s − 16.0·14-s + 13·16-s + 1.45·17-s − 1.41·18-s + 0.458·19-s − 2.55·22-s + 1.25·23-s + 34.0·28-s − 0.742·29-s − 1.07·31-s − 8.48·32-s − 6.17·34-s + 3·36-s − 2.95·37-s − 1.94·38-s + 0.624·41-s + 5.42·44-s − 5.30·46-s + 61/7·49-s + 4.94·53-s − 48.1·56-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{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⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 308.848308.848
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.63922212960.6392221296
L(12)L(\frac12) ≈\approx 0.63922212960.6392221296
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
7C2C_2 (1−5T+pT2)2 ( 1 - 5 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} )
11D4×C2D_4\times C_2 1−2T−16T2+4T3+235T4+4pT5−16p2T6−2p3T7+p4T8 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1−14T2+15pT4−14p2T6+p4T8 1 - 14 T^{2} + 15 p T^{4} - 14 p^{2} T^{6} + p^{4} T^{8}
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 1−2T−23T2+22T3+292T4+22pT5−23p2T6−2p3T7+p4T8 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 22 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−6T+52T2−240T3+1347T4−240pT5+52p2T6−6p3T7+p4T8 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
29D4D_{4} (1+2T+32T2+2pT3+p2T4)2 ( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
31D4×C2D_4\times C_2 1+6T−23T2−18T3+1404T4−18pT5−23p2T6+6p3T7+p4T8 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
37D4×C2D_4\times C_2 1+18T+205T2+1746T3+12036T4+1746pT5+205p2T6+18p3T7+p4T8 1 + 18 T + 205 T^{2} + 1746 T^{3} + 12036 T^{4} + 1746 p T^{5} + 205 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}
41D4D_{4} (1−2T+80T2−2pT3+p2T4)2 ( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−110T2+6291T4−110p2T6+p4T8 1 - 110 T^{2} + 6291 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8}
47C23C_2^3 1+90T2+5891T4+90p2T6+p4T8 1 + 90 T^{2} + 5891 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−36T+642T2−7560T3+64187T4−7560pT5+642p2T6−36p3T7+p4T8 1 - 36 T + 642 T^{2} - 7560 T^{3} + 64187 T^{4} - 7560 p T^{5} + 642 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8}
59D4×C2D_4\times C_2 1−10T−16T2+20T3+4075T4+20pT5−16p2T6−10p3T7+p4T8 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 20 p T^{5} - 16 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
61C22C_2^2 (1+4T−45T2+4pT3+p2T4)2 ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−30T+473T2−5190T3+45540T4−5190pT5+473p2T6−30p3T7+p4T8 1 - 30 T + 473 T^{2} - 5190 T^{3} + 45540 T^{4} - 5190 p T^{5} + 473 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8}
71D4D_{4} (1−2T+116T2−2pT3+p2T4)2 ( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−30T+505T2−6150T3+58596T4−6150pT5+505p2T6−30p3T7+p4T8 1 - 30 T + 505 T^{2} - 6150 T^{3} + 58596 T^{4} - 6150 p T^{5} + 505 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8}
79D4×C2D_4\times C_2 1−6T−23T2+594T3−5604T4+594pT5−23p2T6−6p3T7+p4T8 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 594 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1−20T2+8586T4−20p2T6+p4T8 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
89D4×C2D_4\times C_2 1−6T−4T2+828T3−9525T4+828pT5−4p2T6−6p3T7+p4T8 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 828 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−164T2+13254T4−164p2T6+p4T8 1 - 164 T^{2} + 13254 T^{4} - 164 p^{2} T^{6} + 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

−7.902901532447817267410596539800, −7.71456437367338434598713672274, −7.48052191516048858585644476890, −7.45910403246871476877594419935, −7.39999107185373102339085595692, −6.81795350565640618371458417213, −6.69952447536074903986805879981, −6.64899230199804599491697035486, −5.81052105334269405746371238762, −5.66465800535425033931959532244, −5.43682811367613161335282733701, −5.06017043934034719462328163870, −4.94380827983789894853145636433, −4.88605919926622449298858228661, −4.22608444829518348100613054512, −3.75150188242948291701146128565, −3.66807239622448054706647150160, −3.43695980300812316507625698378, −2.26998119543594401573493276492, −2.24208178343154337055165461172, −2.17877235028455554693608118324, −1.72609792241351778106834425704, −0.975275579198860834757675919719, −0.952338313780787739264269641214, −0.937870288855792845004158091441, 0.937870288855792845004158091441, 0.952338313780787739264269641214, 0.975275579198860834757675919719, 1.72609792241351778106834425704, 2.17877235028455554693608118324, 2.24208178343154337055165461172, 2.26998119543594401573493276492, 3.43695980300812316507625698378, 3.66807239622448054706647150160, 3.75150188242948291701146128565, 4.22608444829518348100613054512, 4.88605919926622449298858228661, 4.94380827983789894853145636433, 5.06017043934034719462328163870, 5.43682811367613161335282733701, 5.66465800535425033931959532244, 5.81052105334269405746371238762, 6.64899230199804599491697035486, 6.69952447536074903986805879981, 6.81795350565640618371458417213, 7.39999107185373102339085595692, 7.45910403246871476877594419935, 7.48052191516048858585644476890, 7.71456437367338434598713672274, 7.902901532447817267410596539800

Graph of the ZZ-function along the critical line