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Properties

Label 8-364e4-1.1-c1e4-0-3
Degree 88
Conductor 1755519001617555190016
Sign 11
Analytic cond. 71.369771.3697
Root an. cond. 1.704861.70486
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  − 3-s + 6·5-s + 2·7-s + 3·9-s − 9·11-s − 6·15-s − 4·17-s − 11·19-s − 2·21-s − 4·23-s + 9·25-s − 2·27-s − 3·29-s − 16·31-s + 9·33-s + 12·35-s − 4·37-s + 10·41-s + 43-s + 18·45-s + 32·47-s + 49-s + 4·51-s − 54·55-s + 11·57-s + 4·59-s + 6·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 2.68·5-s + 0.755·7-s + 9-s − 2.71·11-s − 1.54·15-s − 0.970·17-s − 2.52·19-s − 0.436·21-s − 0.834·23-s + 9/5·25-s − 0.384·27-s − 0.557·29-s − 2.87·31-s + 1.56·33-s + 2.02·35-s − 0.657·37-s + 1.56·41-s + 0.152·43-s + 2.68·45-s + 4.66·47-s + 1/7·49-s + 0.560·51-s − 7.28·55-s + 1.45·57-s + 0.520·59-s + 0.755·63-s + ⋯

Functional equation

Λ(s)=((28⋅74⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅74⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \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: 28⋅74⋅1342^{8} \cdot 7^{4} \cdot 13^{4}
Sign: 11
Analytic conductor: 71.369771.3697
Root analytic conductor: 1.704861.70486
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅74⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.5540516701.554051670
L(12)L(\frac12) ≈\approx 1.5540516701.554051670
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)
bad2 1 1
7C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
13C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
good3D4×C2D_4\times C_2 1+T−2T2−pT3−pT4−p2T5−2p2T6+p3T7+p4T8 1 + T - 2 T^{2} - p T^{3} - p T^{4} - p^{2} T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
5D4D_{4} (1−3T+9T2−3pT3+p2T4)2 ( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
11D4×C2D_4\times C_2 1+9T+42T2+153T3+509T4+153pT5+42p2T6+9p3T7+p4T8 1 + 9 T + 42 T^{2} + 153 T^{3} + 509 T^{4} + 153 p T^{5} + 42 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
17D4×C2D_4\times C_2 1+4T−9T2−36T3+64T4−36pT5−9p2T6+4p3T7+p4T8 1 + 4 T - 9 T^{2} - 36 T^{3} + 64 T^{4} - 36 p T^{5} - 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1+11T+56T2+297T3+1565T4+297pT5+56p2T6+11p3T7+p4T8 1 + 11 T + 56 T^{2} + 297 T^{3} + 1565 T^{4} + 297 p T^{5} + 56 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}
23C22C_2^2 (1+2T−19T2+2pT3+p2T4)2 ( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
29D4×C2D_4\times C_2 1+3T−48T2−3T3+2147T4−3pT5−48p2T6+3p3T7+p4T8 1 + 3 T - 48 T^{2} - 3 T^{3} + 2147 T^{4} - 3 p T^{5} - 48 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
31D4D_{4} (1+8T+65T2+8pT3+p2T4)2 ( 1 + 8 T + 65 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1+4T−10T2−192T3−1285T4−192pT5−10p2T6+4p3T7+p4T8 1 + 4 T - 10 T^{2} - 192 T^{3} - 1285 T^{4} - 192 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
41D4×C2D_4\times C_2 1−10T+6T2−120T3+3055T4−120pT5+6p2T6−10p3T7+p4T8 1 - 10 T + 6 T^{2} - 120 T^{3} + 3055 T^{4} - 120 p T^{5} + 6 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−T−56T2+29T3+1357T4+29pT5−56p2T6−p3T7+p4T8 1 - T - 56 T^{2} + 29 T^{3} + 1357 T^{4} + 29 p T^{5} - 56 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
47D4D_{4} (1−16T+145T2−16pT3+p2T4)2 ( 1 - 16 T + 145 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−11T2+p2T4)2 ( 1 - 11 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−4T−93T2+36T3+7456T4+36pT5−93p2T6−4p3T7+p4T8 1 - 4 T - 93 T^{2} + 36 T^{3} + 7456 T^{4} + 36 p T^{5} - 93 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
61C22C_2^2×\timesC22C_2^2 (1−24T+253T2−24pT3+p2T4)(1+24T+253T2+24pT3+p2T4) ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} )
67C22C_2^2 (1+13T+102T2+13pT3+p2T4)2 ( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2}
71D4×C2D_4\times C_2 1+2T−22T2−232T3−4649T4−232pT5−22p2T6+2p3T7+p4T8 1 + 2 T - 22 T^{2} - 232 T^{3} - 4649 T^{4} - 232 p T^{5} - 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
73D4D_{4} (1−8T+110T2−8pT3+p2T4)2 ( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
83D4D_{4} (1+16T+217T2+16pT3+p2T4)2 ( 1 + 16 T + 217 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1+21T+182T2+1701T3+19911T4+1701pT5+182p2T6+21p3T7+p4T8 1 + 21 T + 182 T^{2} + 1701 T^{3} + 19911 T^{4} + 1701 p T^{5} + 182 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−23T+232T2−2369T3+27487T4−2369pT5+232p2T6−23p3T7+p4T8 1 - 23 T + 232 T^{2} - 2369 T^{3} + 27487 T^{4} - 2369 p T^{5} + 232 p^{2} T^{6} - 23 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

−8.093613534311314808538507129134, −7.998800979577138175609824982962, −7.928344581250233956817722093157, −7.38596674438009953330486377799, −7.25038928674439170479713036974, −7.17992736912262435092148485977, −6.77196669989586514359521555106, −6.26535634696070366813830564331, −6.15004011573173469883792758699, −5.85481695330638953558994276287, −5.65703401036892100952047137196, −5.56754909329035572350907897989, −5.42725490490203668547434590180, −4.97625393284363821588813098299, −4.58464912842067791282364016506, −4.38344948311692887120068685202, −4.08775994471865909359428653116, −3.84243154353666024426038462535, −3.24183421362289111243489981232, −2.45436803452976693604073378195, −2.42955462695472090292290956267, −2.15891319285075305321793199793, −1.87620254324778795540896787179, −1.70663876683441375589242825840, −0.46584609507010885795392899502, 0.46584609507010885795392899502, 1.70663876683441375589242825840, 1.87620254324778795540896787179, 2.15891319285075305321793199793, 2.42955462695472090292290956267, 2.45436803452976693604073378195, 3.24183421362289111243489981232, 3.84243154353666024426038462535, 4.08775994471865909359428653116, 4.38344948311692887120068685202, 4.58464912842067791282364016506, 4.97625393284363821588813098299, 5.42725490490203668547434590180, 5.56754909329035572350907897989, 5.65703401036892100952047137196, 5.85481695330638953558994276287, 6.15004011573173469883792758699, 6.26535634696070366813830564331, 6.77196669989586514359521555106, 7.17992736912262435092148485977, 7.25038928674439170479713036974, 7.38596674438009953330486377799, 7.928344581250233956817722093157, 7.998800979577138175609824982962, 8.093613534311314808538507129134

Graph of the ZZ-function along the critical line