Properties

Label 8-1925e4-1.1-c3e4-0-1
Degree 88
Conductor 1.373×10131.373\times 10^{13}
Sign 11
Analytic cond. 1.66412×1081.66412\times 10^{8}
Root an. cond. 10.657310.6573
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 14·3-s − 4-s − 28·6-s + 28·7-s − 4·8-s + 82·9-s − 44·11-s + 14·12-s − 58·13-s + 56·14-s − 51·16-s − 4·17-s + 164·18-s + 258·19-s − 392·21-s − 88·22-s − 8·23-s + 56·24-s − 116·26-s − 250·27-s − 28·28-s − 396·29-s − 56·31-s − 154·32-s + 616·33-s − 8·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.69·3-s − 1/8·4-s − 1.90·6-s + 1.51·7-s − 0.176·8-s + 3.03·9-s − 1.20·11-s + 0.336·12-s − 1.23·13-s + 1.06·14-s − 0.796·16-s − 0.0570·17-s + 2.14·18-s + 3.11·19-s − 4.07·21-s − 0.852·22-s − 0.0725·23-s + 0.476·24-s − 0.874·26-s − 1.78·27-s − 0.188·28-s − 2.53·29-s − 0.324·31-s − 0.850·32-s + 3.24·33-s − 0.0403·34-s + ⋯

Functional equation

Λ(s)=((5874114)s/2ΓC(s)4L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((5874114)s/2ΓC(s+3/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 58741145^{8} \cdot 7^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 1.66412×1081.66412\times 10^{8}
Root analytic conductor: 10.657310.6573
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 5874114, ( :3/2,3/2,3/2,3/2), 1)(8,\ 5^{8} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad5 1 1
7C1C_1 (1pT)4 ( 1 - p T )^{4}
11C1C_1 (1+pT)4 ( 1 + p T )^{4}
good2C2S4C_2 \wr S_4 1pT+5T2p3T3+p6T4p6T5+5p6T6p10T7+p12T8 1 - p T + 5 T^{2} - p^{3} T^{3} + p^{6} T^{4} - p^{6} T^{5} + 5 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8}
3C2S4C_2 \wr S_4 1+14T+38pT2+698T3+3682T4+698p3T5+38p7T6+14p9T7+p12T8 1 + 14 T + 38 p T^{2} + 698 T^{3} + 3682 T^{4} + 698 p^{3} T^{5} + 38 p^{7} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8}
13C2S4C_2 \wr S_4 1+58T+3862T2+39850T3+2368354T4+39850p3T5+3862p6T6+58p9T7+p12T8 1 + 58 T + 3862 T^{2} + 39850 T^{3} + 2368354 T^{4} + 39850 p^{3} T^{5} + 3862 p^{6} T^{6} + 58 p^{9} T^{7} + p^{12} T^{8}
17C2S4C_2 \wr S_4 1+4T+13466T2198500T3+81336754T4198500p3T5+13466p6T6+4p9T7+p12T8 1 + 4 T + 13466 T^{2} - 198500 T^{3} + 81336754 T^{4} - 198500 p^{3} T^{5} + 13466 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8}
19C2S4C_2 \wr S_4 1258T+2296pT25147154T3+489918366T45147154p3T5+2296p7T6258p9T7+p12T8 1 - 258 T + 2296 p T^{2} - 5147154 T^{3} + 489918366 T^{4} - 5147154 p^{3} T^{5} + 2296 p^{7} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8}
23C2S4C_2 \wr S_4 1+8T+26276T21258456T3+325878550T41258456p3T5+26276p6T6+8p9T7+p12T8 1 + 8 T + 26276 T^{2} - 1258456 T^{3} + 325878550 T^{4} - 1258456 p^{3} T^{5} + 26276 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8}
29C2S4C_2 \wr S_4 1+396T+138428T2+30598020T3+5584930806T4+30598020p3T5+138428p6T6+396p9T7+p12T8 1 + 396 T + 138428 T^{2} + 30598020 T^{3} + 5584930806 T^{4} + 30598020 p^{3} T^{5} + 138428 p^{6} T^{6} + 396 p^{9} T^{7} + p^{12} T^{8}
31C2S4C_2 \wr S_4 1+56T+87274T2+3513992T3+3413701858T4+3513992p3T5+87274p6T6+56p9T7+p12T8 1 + 56 T + 87274 T^{2} + 3513992 T^{3} + 3413701858 T^{4} + 3513992 p^{3} T^{5} + 87274 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8}
37C2S4C_2 \wr S_4 1+84T+139096T2+12117036T3+8970963870T4+12117036p3T5+139096p6T6+84p9T7+p12T8 1 + 84 T + 139096 T^{2} + 12117036 T^{3} + 8970963870 T^{4} + 12117036 p^{3} T^{5} + 139096 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8}
41C2S4C_2 \wr S_4 152T+191978T2+4000964T3+16303189330T4+4000964p3T5+191978p6T652p9T7+p12T8 1 - 52 T + 191978 T^{2} + 4000964 T^{3} + 16303189330 T^{4} + 4000964 p^{3} T^{5} + 191978 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8}
43C2S4C_2 \wr S_4 1+408T+227224T2+65489736T3+24699468414T4+65489736p3T5+227224p6T6+408p9T7+p12T8 1 + 408 T + 227224 T^{2} + 65489736 T^{3} + 24699468414 T^{4} + 65489736 p^{3} T^{5} + 227224 p^{6} T^{6} + 408 p^{9} T^{7} + p^{12} T^{8}
47C2S4C_2 \wr S_4 1+8T+293642T2+14787896T3+39096224482T4+14787896p3T5+293642p6T6+8p9T7+p12T8 1 + 8 T + 293642 T^{2} + 14787896 T^{3} + 39096224482 T^{4} + 14787896 p^{3} T^{5} + 293642 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8}
53C2S4C_2 \wr S_4 1+624T+731456T2+291124560T3+173869150638T4+291124560p3T5+731456p6T6+624p9T7+p12T8 1 + 624 T + 731456 T^{2} + 291124560 T^{3} + 173869150638 T^{4} + 291124560 p^{3} T^{5} + 731456 p^{6} T^{6} + 624 p^{9} T^{7} + p^{12} T^{8}
59C2S4C_2 \wr S_4 1+238T+320090T23740918T3+64718281666T43740918p3T5+320090p6T6+238p9T7+p12T8 1 + 238 T + 320090 T^{2} - 3740918 T^{3} + 64718281666 T^{4} - 3740918 p^{3} T^{5} + 320090 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8}
61C2S4C_2 \wr S_4 1+162T+320206T256573262T3+48989538114T456573262p3T5+320206p6T6+162p9T7+p12T8 1 + 162 T + 320206 T^{2} - 56573262 T^{3} + 48989538114 T^{4} - 56573262 p^{3} T^{5} + 320206 p^{6} T^{6} + 162 p^{9} T^{7} + p^{12} T^{8}
67C2S4C_2 \wr S_4 1+20pT+1030948T2+550745180T3+298359795814T4+550745180p3T5+1030948p6T6+20p10T7+p12T8 1 + 20 p T + 1030948 T^{2} + 550745180 T^{3} + 298359795814 T^{4} + 550745180 p^{3} T^{5} + 1030948 p^{6} T^{6} + 20 p^{10} T^{7} + p^{12} T^{8}
71C2S4C_2 \wr S_4 11788T+2193428T21809946572T3+1240921367286T41809946572p3T5+2193428p6T61788p9T7+p12T8 1 - 1788 T + 2193428 T^{2} - 1809946572 T^{3} + 1240921367286 T^{4} - 1809946572 p^{3} T^{5} + 2193428 p^{6} T^{6} - 1788 p^{9} T^{7} + p^{12} T^{8}
73C2S4C_2 \wr S_4 1+1456T+1167226T2+565357096T3+283420832722T4+565357096p3T5+1167226p6T6+1456p9T7+p12T8 1 + 1456 T + 1167226 T^{2} + 565357096 T^{3} + 283420832722 T^{4} + 565357096 p^{3} T^{5} + 1167226 p^{6} T^{6} + 1456 p^{9} T^{7} + p^{12} T^{8}
79C2S4C_2 \wr S_4 1+1324T+2043976T2+1611726940T3+1469807561518T4+1611726940p3T5+2043976p6T6+1324p9T7+p12T8 1 + 1324 T + 2043976 T^{2} + 1611726940 T^{3} + 1469807561518 T^{4} + 1611726940 p^{3} T^{5} + 2043976 p^{6} T^{6} + 1324 p^{9} T^{7} + p^{12} T^{8}
83C2S4C_2 \wr S_4 1+450T+1903832T2+631295250T3+1545243120894T4+631295250p3T5+1903832p6T6+450p9T7+p12T8 1 + 450 T + 1903832 T^{2} + 631295250 T^{3} + 1545243120894 T^{4} + 631295250 p^{3} T^{5} + 1903832 p^{6} T^{6} + 450 p^{9} T^{7} + p^{12} T^{8}
89C2S4C_2 \wr S_4 1+3072T+5688284T2+7201250304T3+6916861622502T4+7201250304p3T5+5688284p6T6+3072p9T7+p12T8 1 + 3072 T + 5688284 T^{2} + 7201250304 T^{3} + 6916861622502 T^{4} + 7201250304 p^{3} T^{5} + 5688284 p^{6} T^{6} + 3072 p^{9} T^{7} + p^{12} T^{8}
97C2S4C_2 \wr S_4 1652T+1754332T21157057716T3+2404953539782T41157057716p3T5+1754332p6T6652p9T7+p12T8 1 - 652 T + 1754332 T^{2} - 1157057716 T^{3} + 2404953539782 T^{4} - 1157057716 p^{3} T^{5} + 1754332 p^{6} T^{6} - 652 p^{9} T^{7} + p^{12} T^{8}
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   L(s)=p j=18(1αj,pps)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

−6.56054503816477307642447452244, −6.27541295474257218050246558202, −5.87227488915221248872508984693, −5.70466952102789921708955412297, −5.60357201226020354468648835936, −5.43650652225122065767353265206, −5.42177629686587462295808339678, −5.21927231854748104596508944415, −5.11328117649201450051913517960, −4.76458300511517225038525932865, −4.48739823661913007015850299937, −4.36955910148303407806303780605, −4.32882645216477692558027505337, −3.96348087064170576557546987920, −3.57638184949549429285793217821, −3.24534135518849802573170881736, −2.95993286772760325130379001315, −2.87902316241498795960511413372, −2.73513262057433906995558452778, −2.05698249644274196921216776357, −1.74991222174700817016838783528, −1.73281192446457696294183096003, −1.48085365700395338844793837556, −0.904184601321824200233560170188, −0.896430905870101369595148155590, 0, 0, 0, 0, 0.896430905870101369595148155590, 0.904184601321824200233560170188, 1.48085365700395338844793837556, 1.73281192446457696294183096003, 1.74991222174700817016838783528, 2.05698249644274196921216776357, 2.73513262057433906995558452778, 2.87902316241498795960511413372, 2.95993286772760325130379001315, 3.24534135518849802573170881736, 3.57638184949549429285793217821, 3.96348087064170576557546987920, 4.32882645216477692558027505337, 4.36955910148303407806303780605, 4.48739823661913007015850299937, 4.76458300511517225038525932865, 5.11328117649201450051913517960, 5.21927231854748104596508944415, 5.42177629686587462295808339678, 5.43650652225122065767353265206, 5.60357201226020354468648835936, 5.70466952102789921708955412297, 5.87227488915221248872508984693, 6.27541295474257218050246558202, 6.56054503816477307642447452244

Graph of the ZZ-function along the critical line