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Properties

Label 8-525e4-1.1-c1e4-0-15
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·4-s + 2·7-s + 9-s − 4·11-s − 4·12-s + 12·13-s + 4·16-s − 4·17-s + 2·19-s − 4·21-s − 4·23-s + 2·27-s + 4·28-s + 16·29-s + 6·31-s + 8·33-s + 2·36-s − 14·37-s − 24·39-s − 8·41-s + 36·43-s − 8·44-s − 4·47-s − 8·48-s + 7·49-s + 8·51-s + ⋯
L(s)  = 1  − 1.15·3-s + 4-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.15·12-s + 3.32·13-s + 16-s − 0.970·17-s + 0.458·19-s − 0.872·21-s − 0.834·23-s + 0.384·27-s + 0.755·28-s + 2.97·29-s + 1.07·31-s + 1.39·33-s + 1/3·36-s − 2.30·37-s − 3.84·39-s − 1.24·41-s + 5.48·43-s − 1.20·44-s − 0.583·47-s − 1.15·48-s + 49-s + 1.12·51-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 3.2416819743.241681974
L(12)L(\frac12) ≈\approx 3.2416819743.241681974
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)
bad3C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
5 1 1
7C22C_2^2 1−2T−3T2−2pT3+p2T4 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4}
good2C2C_2×\timesC22C_2^2 (1−pT2)2(1+pT2+p2T4) ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} )
11D4×C2D_4\times C_2 1+4T−8T2+8T3+279T4+8pT5−8p2T6+4p3T7+p4T8 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
13D4D_{4} (1−6T+33T2−6pT3+p2T4)2 ( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
17D4×C2D_4\times C_2 1+4T−4T2−56T3−161T4−56pT5−4p2T6+4p3T7+p4T8 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−2T−3T2+62T3−388T4+62pT5−3p2T6−2p3T7+p4T8 1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+4T−16T2−56T3+127T4−56pT5−16p2T6+4p3T7+p4T8 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 56 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
29D4D_{4} (1−8T+56T2−8pT3+p2T4)2 ( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
31D4×C2D_4\times C_2 1−6T−27T2−6T3+1892T4−6pT5−27p2T6−6p3T7+p4T8 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 6 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
37D4×C2D_4\times C_2 1+14T+75T2+658T3+6020T4+658pT5+75p2T6+14p3T7+p4T8 1 + 14 T + 75 T^{2} + 658 T^{3} + 6020 T^{4} + 658 p T^{5} + 75 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
41D4D_{4} (1+4T+68T2+4pT3+p2T4)2 ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
43D4D_{4} (1−18T+165T2−18pT3+p2T4)2 ( 1 - 18 T + 165 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}
47D4×C2D_4\times C_2 1+4T+46T2−496T3−2061T4−496pT5+46p2T6+4p3T7+p4T8 1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 496 p T^{5} + 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−8T−50T2−64T3+6795T4−64pT5−50p2T6−8p3T7+p4T8 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
59C23C_2^3 1−116T2+9975T4−116p2T6+p4T8 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+8T−2T2−448T3−3269T4−448pT5−2p2T6+8p3T7+p4T8 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 448 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1+2T+31T2−322T3−4028T4−322pT5+31p2T6+2p3T7+p4T8 1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 322 p T^{5} + 31 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
71D4D_{4} (1+4T+144T2+4pT3+p2T4)2 ( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1+10T−21T2−250T3+2012T4−250pT5−21p2T6+10p3T7+p4T8 1 + 10 T - 21 T^{2} - 250 T^{3} + 2012 T^{4} - 250 p T^{5} - 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
79D4×C2D_4\times C_2 1+2T−123T2−62T3+9572T4−62pT5−123p2T6+2p3T7+p4T8 1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 62 p T^{5} - 123 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
83D4D_{4} (1+8T+180T2+8pT3+p2T4)2 ( 1 + 8 T + 180 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1−16T+32T2−736T3+19471T4−736pT5+32p2T6−16p3T7+p4T8 1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 736 p T^{5} + 32 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
97D4D_{4} (1−16T+250T2−16pT3+p2T4)2 ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} 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

−7.59680816073005120190443971702, −7.50915357451069729783977342355, −7.44011287278583518695134941053, −7.05558052780846275197310253356, −7.00456498950766796807564399515, −6.46368048800085617685875886813, −6.17991254443848803928791093872, −5.95355349157480781183110511691, −5.92277023270848668495261901851, −5.88870260812422690972763569435, −5.75556878130513889407475945658, −4.90219962096394044195593052436, −4.76185966475079827734485526396, −4.67672434575019028170946460263, −4.62101962538669186801178569230, −3.80927029638126628561973823840, −3.71232974096582585921858578820, −3.44207219437497943693608926365, −3.12751126942958120730267843995, −2.56817056252267954857911553392, −2.35162858068350172181695472222, −2.08236511203292890354092003344, −1.42435431579097682436193401469, −0.940784216866663612608743471700, −0.839952529568944702235911553116, 0.839952529568944702235911553116, 0.940784216866663612608743471700, 1.42435431579097682436193401469, 2.08236511203292890354092003344, 2.35162858068350172181695472222, 2.56817056252267954857911553392, 3.12751126942958120730267843995, 3.44207219437497943693608926365, 3.71232974096582585921858578820, 3.80927029638126628561973823840, 4.62101962538669186801178569230, 4.67672434575019028170946460263, 4.76185966475079827734485526396, 4.90219962096394044195593052436, 5.75556878130513889407475945658, 5.88870260812422690972763569435, 5.92277023270848668495261901851, 5.95355349157480781183110511691, 6.17991254443848803928791093872, 6.46368048800085617685875886813, 7.00456498950766796807564399515, 7.05558052780846275197310253356, 7.44011287278583518695134941053, 7.50915357451069729783977342355, 7.59680816073005120190443971702

Graph of the ZZ-function along the critical line