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Properties

Label 8-9702e4-1.1-c1e4-0-3
Degree 88
Conductor 8.860×10158.860\times 10^{15}
Sign 11
Analytic cond. 3.60208×1073.60208\times 10^{7}
Root an. cond. 8.801758.80175
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  + 4·2-s + 10·4-s + 20·8-s + 4·11-s + 35·16-s + 16·22-s + 8·23-s − 4·25-s + 16·31-s + 56·32-s + 8·37-s + 8·43-s + 40·44-s + 32·46-s − 16·47-s − 16·50-s − 8·53-s − 32·59-s + 16·61-s + 64·62-s + 84·64-s + 16·67-s + 32·74-s + 16·79-s + 32·86-s + 80·88-s − 16·89-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 7.07·8-s + 1.20·11-s + 35/4·16-s + 3.41·22-s + 1.66·23-s − 4/5·25-s + 2.87·31-s + 9.89·32-s + 1.31·37-s + 1.21·43-s + 6.03·44-s + 4.71·46-s − 2.33·47-s − 2.26·50-s − 1.09·53-s − 4.16·59-s + 2.04·61-s + 8.12·62-s + 21/2·64-s + 1.95·67-s + 3.71·74-s + 1.80·79-s + 3.45·86-s + 8.52·88-s − 1.69·89-s + ⋯

Functional equation

Λ(s)=((24⋅38⋅78⋅114)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅38⋅78⋅114)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅38⋅78⋅1142^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}
Sign: 11
Analytic conductor: 3.60208×1073.60208\times 10^{7}
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅38⋅78⋅114, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 109.6123430109.6123430
L(12)L(\frac12) ≈\approx 109.6123430109.6123430
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)
bad2C1C_1 (1−T)4 ( 1 - T )^{4}
3 1 1
7 1 1
11C1C_1 (1−T)4 ( 1 - T )^{4}
good5C22≀C2C_2^2 \wr C_2 1+4T2+46T4+4p2T6+p4T8 1 + 4 T^{2} + 46 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}
13C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T2−32T3+242T4−32pT5+16p2T6+p4T8 1 + 16 T^{2} - 32 T^{3} + 242 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8}
17C2≀C2≀C2C_2 \wr C_2\wr C_2 1+36T2−32T3+638T4−32pT5+36p2T6+p4T8 1 + 36 T^{2} - 32 T^{3} + 638 T^{4} - 32 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8}
19C2≀C2≀C2C_2 \wr C_2\wr C_2 1+40T2+80T3+794T4+80pT5+40p2T6+p4T8 1 + 40 T^{2} + 80 T^{3} + 794 T^{4} + 80 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8}
23C2≀C2≀C2C_2 \wr C_2\wr C_2 1−8T+36T2+56T3−570T4+56pT5+36p2T6−8p3T7+p4T8 1 - 8 T + 36 T^{2} + 56 T^{3} - 570 T^{4} + 56 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
29C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
31C2≀C2≀C2C_2 \wr C_2\wr C_2 1−16T+200T2−1568T3+10378T4−1568pT5+200p2T6−16p3T7+p4T8 1 - 16 T + 200 T^{2} - 1568 T^{3} + 10378 T^{4} - 1568 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
37C2≀C2≀C2C_2 \wr C_2\wr C_2 1−8T+76T2−408T3+2486T4−408pT5+76p2T6−8p3T7+p4T8 1 - 8 T + 76 T^{2} - 408 T^{3} + 2486 T^{4} - 408 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
41C2≀C2≀C2C_2 \wr C_2\wr C_2 1+132T2+32T3+7454T4+32pT5+132p2T6+p4T8 1 + 132 T^{2} + 32 T^{3} + 7454 T^{4} + 32 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8}
43C2≀C2≀C2C_2 \wr C_2\wr C_2 1−8T+116T2−424T3+5110T4−424pT5+116p2T6−8p3T7+p4T8 1 - 8 T + 116 T^{2} - 424 T^{3} + 5110 T^{4} - 424 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
47C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T+248T2+2304T3+18890T4+2304pT5+248p2T6+16p3T7+p4T8 1 + 16 T + 248 T^{2} + 2304 T^{3} + 18890 T^{4} + 2304 p T^{5} + 248 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
53C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+140T2+1048T3+9334T4+1048pT5+140p2T6+8p3T7+p4T8 1 + 8 T + 140 T^{2} + 1048 T^{3} + 9334 T^{4} + 1048 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
59D4D_{4} (1+16T+174T2+16pT3+p2T4)2 ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
61C2≀C2≀C2C_2 \wr C_2\wr C_2 1−16T+304T2−2864T3+29362T4−2864pT5+304p2T6−16p3T7+p4T8 1 - 16 T + 304 T^{2} - 2864 T^{3} + 29362 T^{4} - 2864 p T^{5} + 304 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
67C2≀C2≀C2C_2 \wr C_2\wr C_2 1−16T+124T2−1104T3+11510T4−1104pT5+124p2T6−16p3T7+p4T8 1 - 16 T + 124 T^{2} - 1104 T^{3} + 11510 T^{4} - 1104 p T^{5} + 124 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
71C2≀C2≀C2C_2 \wr C_2\wr C_2 1+140T2+256T3+12422T4+256pT5+140p2T6+p4T8 1 + 140 T^{2} + 256 T^{3} + 12422 T^{4} + 256 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8}
73C2≀C2≀C2C_2 \wr C_2\wr C_2 1+116T2−448T3+6462T4−448pT5+116p2T6+p4T8 1 + 116 T^{2} - 448 T^{3} + 6462 T^{4} - 448 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8}
79C2≀C2≀C2C_2 \wr C_2\wr C_2 1−16T+236T2−2448T3+24582T4−2448pT5+236p2T6−16p3T7+p4T8 1 - 16 T + 236 T^{2} - 2448 T^{3} + 24582 T^{4} - 2448 p T^{5} + 236 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
83C2≀C2≀C2C_2 \wr C_2\wr C_2 1+280T2+48T3+32794T4+48pT5+280p2T6+p4T8 1 + 280 T^{2} + 48 T^{3} + 32794 T^{4} + 48 p T^{5} + 280 p^{2} T^{6} + p^{4} T^{8}
89C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T+160T2+1904T3+24642T4+1904pT5+160p2T6+16p3T7+p4T8 1 + 16 T + 160 T^{2} + 1904 T^{3} + 24642 T^{4} + 1904 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
97C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T+352T2+3984T3+51458T4+3984pT5+352p2T6+16p3T7+p4T8 1 + 16 T + 352 T^{2} + 3984 T^{3} + 51458 T^{4} + 3984 p T^{5} + 352 p^{2} T^{6} + 16 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

−5.48552228788439949160136406502, −5.24148545148005391033831629515, −4.94508857249820285055012393814, −4.70234835600549987876508252077, −4.62522315205475673158084260645, −4.45145323258622128395809210767, −4.31045226840597124346438033659, −4.17025833486356583580176413556, −4.04060077385134389081472125588, −3.87970392220113667890541132211, −3.41278598763777026428041312173, −3.33292651936476729224546006944, −3.23325929674411569881889109414, −2.97486351769955746842570513503, −2.80704125945089637808992115212, −2.73387894166343282327866351614, −2.59993280436734521194587103711, −2.00334853676016076889840769628, −1.90018564425422211803507168312, −1.83309377357171726648245547131, −1.56558271460816864309062141549, −1.25051474787283699224361739118, −0.925643581893000526428842730780, −0.59569537409131425080546109122, −0.59201217388398950288287038429, 0.59201217388398950288287038429, 0.59569537409131425080546109122, 0.925643581893000526428842730780, 1.25051474787283699224361739118, 1.56558271460816864309062141549, 1.83309377357171726648245547131, 1.90018564425422211803507168312, 2.00334853676016076889840769628, 2.59993280436734521194587103711, 2.73387894166343282327866351614, 2.80704125945089637808992115212, 2.97486351769955746842570513503, 3.23325929674411569881889109414, 3.33292651936476729224546006944, 3.41278598763777026428041312173, 3.87970392220113667890541132211, 4.04060077385134389081472125588, 4.17025833486356583580176413556, 4.31045226840597124346438033659, 4.45145323258622128395809210767, 4.62522315205475673158084260645, 4.70234835600549987876508252077, 4.94508857249820285055012393814, 5.24148545148005391033831629515, 5.48552228788439949160136406502

Graph of the ZZ-function along the critical line