Properties

Label 8-650e4-1.1-c3e4-0-11
Degree 88
Conductor 178506250000178506250000
Sign 11
Analytic cond. 2.16330×1062.16330\times 10^{6}
Root an. cond. 6.192836.19283
Motivic weight 33
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  + 8·2-s + 40·4-s + 56·7-s + 160·8-s + 5·9-s − 49·11-s + 52·13-s + 448·14-s + 560·16-s + 27·17-s + 40·18-s + 51·19-s − 392·22-s − 9·23-s + 416·26-s − 48·27-s + 2.24e3·28-s − 161·29-s + 38·31-s + 1.79e3·32-s + 216·34-s + 200·36-s + 16·37-s + 408·38-s + 547·41-s + 609·43-s − 1.96e3·44-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 3.02·7-s + 7.07·8-s + 5/27·9-s − 1.34·11-s + 1.10·13-s + 8.55·14-s + 35/4·16-s + 0.385·17-s + 0.523·18-s + 0.615·19-s − 3.79·22-s − 0.0815·23-s + 3.13·26-s − 0.342·27-s + 15.1·28-s − 1.03·29-s + 0.220·31-s + 9.89·32-s + 1.08·34-s + 0.925·36-s + 0.0710·37-s + 1.74·38-s + 2.08·41-s + 2.15·43-s − 6.71·44-s + ⋯

Functional equation

Λ(s)=((2458134)s/2ΓC(s)4L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((2458134)s/2ΓC(s+3/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24581342^{4} \cdot 5^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 2.16330×1062.16330\times 10^{6}
Root analytic conductor: 6.192836.19283
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 2458134, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) \approx 128.1454744128.1454744
L(12)L(\frac12) \approx 128.1454744128.1454744
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)
bad2C1C_1 (1pT)4 ( 1 - p T )^{4}
5 1 1
13C1C_1 (1pT)4 ( 1 - p T )^{4}
good3C2S4C_2 \wr S_4 15T2+16pT3128T4+16p4T55p6T6+p12T8 1 - 5 T^{2} + 16 p T^{3} - 128 T^{4} + 16 p^{4} T^{5} - 5 p^{6} T^{6} + p^{12} T^{8}
7C2S4C_2 \wr S_4 18pT+2127T256354T3+1179224T456354p3T5+2127p6T68p10T7+p12T8 1 - 8 p T + 2127 T^{2} - 56354 T^{3} + 1179224 T^{4} - 56354 p^{3} T^{5} + 2127 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8}
11C2S4C_2 \wr S_4 1+49T+3391T2+138994T3+5643870T4+138994p3T5+3391p6T6+49p9T7+p12T8 1 + 49 T + 3391 T^{2} + 138994 T^{3} + 5643870 T^{4} + 138994 p^{3} T^{5} + 3391 p^{6} T^{6} + 49 p^{9} T^{7} + p^{12} T^{8}
17C2S4C_2 \wr S_4 127T+2197T222226T3+35190538T422226p3T5+2197p6T627p9T7+p12T8 1 - 27 T + 2197 T^{2} - 22226 T^{3} + 35190538 T^{4} - 22226 p^{3} T^{5} + 2197 p^{6} T^{6} - 27 p^{9} T^{7} + p^{12} T^{8}
19C2S4C_2 \wr S_4 151T+56p2T21144575T3+184346126T41144575p3T5+56p8T651p9T7+p12T8 1 - 51 T + 56 p^{2} T^{2} - 1144575 T^{3} + 184346126 T^{4} - 1144575 p^{3} T^{5} + 56 p^{8} T^{6} - 51 p^{9} T^{7} + p^{12} T^{8}
23C2S4C_2 \wr S_4 1+9T+19148T21050327T3+211262118T41050327p3T5+19148p6T6+9p9T7+p12T8 1 + 9 T + 19148 T^{2} - 1050327 T^{3} + 211262118 T^{4} - 1050327 p^{3} T^{5} + 19148 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8}
29C2S4C_2 \wr S_4 1+161T+62851T2+8580410T3+2244057936T4+8580410p3T5+62851p6T6+161p9T7+p12T8 1 + 161 T + 62851 T^{2} + 8580410 T^{3} + 2244057936 T^{4} + 8580410 p^{3} T^{5} + 62851 p^{6} T^{6} + 161 p^{9} T^{7} + p^{12} T^{8}
31C2S4C_2 \wr S_4 138T+82751T22524314T3+3420131180T42524314p3T5+82751p6T638p9T7+p12T8 1 - 38 T + 82751 T^{2} - 2524314 T^{3} + 3420131180 T^{4} - 2524314 p^{3} T^{5} + 82751 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8}
37C2S4C_2 \wr S_4 116T+92424T2+3890912T3+4714532798T4+3890912p3T5+92424p6T616p9T7+p12T8 1 - 16 T + 92424 T^{2} + 3890912 T^{3} + 4714532798 T^{4} + 3890912 p^{3} T^{5} + 92424 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8}
41C2S4C_2 \wr S_4 1547T+282444T278101601T3+24924328566T478101601p3T5+282444p6T6547p9T7+p12T8 1 - 547 T + 282444 T^{2} - 78101601 T^{3} + 24924328566 T^{4} - 78101601 p^{3} T^{5} + 282444 p^{6} T^{6} - 547 p^{9} T^{7} + p^{12} T^{8}
43C2S4C_2 \wr S_4 1609T+210408T227097781T3+3443560014T427097781p3T5+210408p6T6609p9T7+p12T8 1 - 609 T + 210408 T^{2} - 27097781 T^{3} + 3443560014 T^{4} - 27097781 p^{3} T^{5} + 210408 p^{6} T^{6} - 609 p^{9} T^{7} + p^{12} T^{8}
47C2S4C_2 \wr S_4 1+254T+329955T2+78532974T3+46994253256T4+78532974p3T5+329955p6T6+254p9T7+p12T8 1 + 254 T + 329955 T^{2} + 78532974 T^{3} + 46994253256 T^{4} + 78532974 p^{3} T^{5} + 329955 p^{6} T^{6} + 254 p^{9} T^{7} + p^{12} T^{8}
53C2S4C_2 \wr S_4 1961T+826473T2439333422T3+201643806858T4439333422p3T5+826473p6T6961p9T7+p12T8 1 - 961 T + 826473 T^{2} - 439333422 T^{3} + 201643806858 T^{4} - 439333422 p^{3} T^{5} + 826473 p^{6} T^{6} - 961 p^{9} T^{7} + p^{12} T^{8}
59C2S4C_2 \wr S_4 1+230T+570243T2+184187238T3+147920246352T4+184187238p3T5+570243p6T6+230p9T7+p12T8 1 + 230 T + 570243 T^{2} + 184187238 T^{3} + 147920246352 T^{4} + 184187238 p^{3} T^{5} + 570243 p^{6} T^{6} + 230 p^{9} T^{7} + p^{12} T^{8}
61C2S4C_2 \wr S_4 11393T+1287731T2871191462T3+481233034676T4871191462p3T5+1287731p6T61393p9T7+p12T8 1 - 1393 T + 1287731 T^{2} - 871191462 T^{3} + 481233034676 T^{4} - 871191462 p^{3} T^{5} + 1287731 p^{6} T^{6} - 1393 p^{9} T^{7} + p^{12} T^{8}
67C2S4C_2 \wr S_4 1579T+478955T266957006T3+73599298802T466957006p3T5+478955p6T6579p9T7+p12T8 1 - 579 T + 478955 T^{2} - 66957006 T^{3} + 73599298802 T^{4} - 66957006 p^{3} T^{5} + 478955 p^{6} T^{6} - 579 p^{9} T^{7} + p^{12} T^{8}
71C2S4C_2 \wr S_4 1680T+1526320T2718747160T3+834788988798T4718747160p3T5+1526320p6T6680p9T7+p12T8 1 - 680 T + 1526320 T^{2} - 718747160 T^{3} + 834788988798 T^{4} - 718747160 p^{3} T^{5} + 1526320 p^{6} T^{6} - 680 p^{9} T^{7} + p^{12} T^{8}
73C2S4C_2 \wr S_4 11321T+1890170T21517221359T3+1160323729322T41517221359p3T5+1890170p6T61321p9T7+p12T8 1 - 1321 T + 1890170 T^{2} - 1517221359 T^{3} + 1160323729322 T^{4} - 1517221359 p^{3} T^{5} + 1890170 p^{6} T^{6} - 1321 p^{9} T^{7} + p^{12} T^{8}
79C2S4C_2 \wr S_4 1+55T+544628T29199713T3+329780786422T49199713p3T5+544628p6T6+55p9T7+p12T8 1 + 55 T + 544628 T^{2} - 9199713 T^{3} + 329780786422 T^{4} - 9199713 p^{3} T^{5} + 544628 p^{6} T^{6} + 55 p^{9} T^{7} + p^{12} T^{8}
83C2S4C_2 \wr S_4 12733T+4979865T25812678818T3+5207004649132T45812678818p3T5+4979865p6T62733p9T7+p12T8 1 - 2733 T + 4979865 T^{2} - 5812678818 T^{3} + 5207004649132 T^{4} - 5812678818 p^{3} T^{5} + 4979865 p^{6} T^{6} - 2733 p^{9} T^{7} + p^{12} T^{8}
89C2S4C_2 \wr S_4 1+1233T+2665102T2+2131465291T3+2654755754554T4+2131465291p3T5+2665102p6T6+1233p9T7+p12T8 1 + 1233 T + 2665102 T^{2} + 2131465291 T^{3} + 2654755754554 T^{4} + 2131465291 p^{3} T^{5} + 2665102 p^{6} T^{6} + 1233 p^{9} T^{7} + p^{12} T^{8}
97C2S4C_2 \wr S_4 11250T+2495976T21844678942T3+2573508001070T41844678942p3T5+2495976p6T61250p9T7+p12T8 1 - 1250 T + 2495976 T^{2} - 1844678942 T^{3} + 2573508001070 T^{4} - 1844678942 p^{3} T^{5} + 2495976 p^{6} T^{6} - 1250 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

−7.17974575182505976891922135176, −6.82030525931740493629307419425, −6.49500516102182579860576941327, −6.44514597265337479662479540840, −6.01663987694777599169674614651, −5.68531253569788687703482767345, −5.63790132492773667237855687565, −5.34631264018484191107742989911, −5.24181917168181349055555506252, −4.92368756401968252689892952692, −4.88549531679258987808338625536, −4.41379863714528606624197143969, −4.36595360976470844219781337698, −3.99830065172604267297895393791, −3.62918893561899683872168337741, −3.53351675285351490675953357109, −3.42830441525640553306252229339, −2.66335058184802715694427739823, −2.39552003818372709361648200977, −2.25912339245938537004282696603, −2.10490838248663253492119317506, −1.71305215942496138245145684718, −1.17554651769401414081471723671, −0.883822737783078875389563975974, −0.73932665550596750286870195681, 0.73932665550596750286870195681, 0.883822737783078875389563975974, 1.17554651769401414081471723671, 1.71305215942496138245145684718, 2.10490838248663253492119317506, 2.25912339245938537004282696603, 2.39552003818372709361648200977, 2.66335058184802715694427739823, 3.42830441525640553306252229339, 3.53351675285351490675953357109, 3.62918893561899683872168337741, 3.99830065172604267297895393791, 4.36595360976470844219781337698, 4.41379863714528606624197143969, 4.88549531679258987808338625536, 4.92368756401968252689892952692, 5.24181917168181349055555506252, 5.34631264018484191107742989911, 5.63790132492773667237855687565, 5.68531253569788687703482767345, 6.01663987694777599169674614651, 6.44514597265337479662479540840, 6.49500516102182579860576941327, 6.82030525931740493629307419425, 7.17974575182505976891922135176

Graph of the ZZ-function along the critical line