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Properties

Label 8-1904e4-1.1-c1e4-0-4
Degree 88
Conductor 1.314×10131.314\times 10^{13}
Sign 11
Analytic cond. 53428.853428.8
Root an. cond. 3.899163.89916
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s + 4·7-s + 2·9-s − 4·11-s − 8·13-s + 3·15-s + 4·17-s − 14·19-s − 12·21-s − 8·23-s − 4·25-s + 2·27-s + 4·29-s − 5·31-s + 12·33-s − 4·35-s − 4·37-s + 24·39-s − 7·41-s − 19·43-s − 2·45-s − 8·47-s + 10·49-s − 12·51-s + 5·53-s + 4·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 1.51·7-s + 2/3·9-s − 1.20·11-s − 2.21·13-s + 0.774·15-s + 0.970·17-s − 3.21·19-s − 2.61·21-s − 1.66·23-s − 4/5·25-s + 0.384·27-s + 0.742·29-s − 0.898·31-s + 2.08·33-s − 0.676·35-s − 0.657·37-s + 3.84·39-s − 1.09·41-s − 2.89·43-s − 0.298·45-s − 1.16·47-s + 10/7·49-s − 1.68·51-s + 0.686·53-s + 0.539·55-s + ⋯

Functional equation

Λ(s)=((216⋅74⋅174)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅74⋅174)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅74⋅1742^{16} \cdot 7^{4} \cdot 17^{4}
Sign: 11
Analytic conductor: 53428.853428.8
Root analytic conductor: 3.899163.89916
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 216⋅74⋅174, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7C1C_1 (1−T)4 ( 1 - T )^{4}
17C1C_1 (1−T)4 ( 1 - T )^{4}
good3(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+pT+7T2+13T3+28T4+13pT5+7p2T6+p4T7+p4T8 1 + p T + 7 T^{2} + 13 T^{3} + 28 T^{4} + 13 p T^{5} + 7 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}
5C2≀C2≀C2C_2 \wr C_2\wr C_2 1+T+pT2−7T3+4T4−7pT5+p3T6+p3T7+p4T8 1 + T + p T^{2} - 7 T^{3} + 4 T^{4} - 7 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8}
11D4D_{4} (1+2T+18T2+2pT3+p2T4)2 ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
13D4D_{4} (1+4T+10T2+4pT3+p2T4)2 ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
19C2≀C2≀C2C_2 \wr C_2\wr C_2 1+14T+116T2+710T3+3510T4+710pT5+116p2T6+14p3T7+p4T8 1 + 14 T + 116 T^{2} + 710 T^{3} + 3510 T^{4} + 710 p T^{5} + 116 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
23C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+72T2+328T3+1998T4+328pT5+72p2T6+8p3T7+p4T8 1 + 8 T + 72 T^{2} + 328 T^{3} + 1998 T^{4} + 328 p T^{5} + 72 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
29C2≀C2≀C2C_2 \wr C_2\wr C_2 1−4T+48T2−364T3+1118T4−364pT5+48p2T6−4p3T7+p4T8 1 - 4 T + 48 T^{2} - 364 T^{3} + 1118 T^{4} - 364 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
31C2≀C2≀C2C_2 \wr C_2\wr C_2 1+5T+39T2−35T3+96T4−35pT5+39p2T6+5p3T7+p4T8 1 + 5 T + 39 T^{2} - 35 T^{3} + 96 T^{4} - 35 p T^{5} + 39 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
37C2≀C2≀C2C_2 \wr C_2\wr C_2 1+4T+80T2+140T3+2878T4+140pT5+80p2T6+4p3T7+p4T8 1 + 4 T + 80 T^{2} + 140 T^{3} + 2878 T^{4} + 140 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
41C2≀C2≀C2C_2 \wr C_2\wr C_2 1+7T+99T2+625T3+5720T4+625pT5+99p2T6+7p3T7+p4T8 1 + 7 T + 99 T^{2} + 625 T^{3} + 5720 T^{4} + 625 p T^{5} + 99 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}
43C2≀C2≀C2C_2 \wr C_2\wr C_2 1+19T+237T2+2183T3+16508T4+2183pT5+237p2T6+19p3T7+p4T8 1 + 19 T + 237 T^{2} + 2183 T^{3} + 16508 T^{4} + 2183 p T^{5} + 237 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8}
47C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+76T2+616T3+5542T4+616pT5+76p2T6+8p3T7+p4T8 1 + 8 T + 76 T^{2} + 616 T^{3} + 5542 T^{4} + 616 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
53C2≀C2≀C2C_2 \wr C_2\wr C_2 1−5T+111T2−575T3+6632T4−575pT5+111p2T6−5p3T7+p4T8 1 - 5 T + 111 T^{2} - 575 T^{3} + 6632 T^{4} - 575 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
59C2≀C2≀C2C_2 \wr C_2\wr C_2 1+192T2−80T3+15758T4−80pT5+192p2T6+p4T8 1 + 192 T^{2} - 80 T^{3} + 15758 T^{4} - 80 p T^{5} + 192 p^{2} T^{6} + p^{4} T^{8}
61C2≀C2≀C2C_2 \wr C_2\wr C_2 1+23T+357T2+3847T3+34148T4+3847pT5+357p2T6+23p3T7+p4T8 1 + 23 T + 357 T^{2} + 3847 T^{3} + 34148 T^{4} + 3847 p T^{5} + 357 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8}
67C2≀C2≀C2C_2 \wr C_2\wr C_2 1+15T+239T2+2555T3+22872T4+2555pT5+239p2T6+15p3T7+p4T8 1 + 15 T + 239 T^{2} + 2555 T^{3} + 22872 T^{4} + 2555 p T^{5} + 239 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}
71C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2T+152T2+1018T3+10798T4+1018pT5+152p2T6+2p3T7+p4T8 1 + 2 T + 152 T^{2} + 1018 T^{3} + 10798 T^{4} + 1018 p T^{5} + 152 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
73C2≀C2≀C2C_2 \wr C_2\wr C_2 1+5T+91T2+155T3+2872T4+155pT5+91p2T6+5p3T7+p4T8 1 + 5 T + 91 T^{2} + 155 T^{3} + 2872 T^{4} + 155 p T^{5} + 91 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
79C2≀C2≀C2C_2 \wr C_2\wr C_2 1−24T+396T2−4920T3+49830T4−4920pT5+396p2T6−24p3T7+p4T8 1 - 24 T + 396 T^{2} - 4920 T^{3} + 49830 T^{4} - 4920 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
83C2≀C2≀C2C_2 \wr C_2\wr C_2 1+10T+212T2+1890T3+25814T4+1890pT5+212p2T6+10p3T7+p4T8 1 + 10 T + 212 T^{2} + 1890 T^{3} + 25814 T^{4} + 1890 p T^{5} + 212 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
89C2≀C2≀C2C_2 \wr C_2\wr C_2 1+16T+136T2+1280T3+16110T4+1280pT5+136p2T6+16p3T7+p4T8 1 + 16 T + 136 T^{2} + 1280 T^{3} + 16110 T^{4} + 1280 p T^{5} + 136 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
97C2≀C2≀C2C_2 \wr C_2\wr C_2 1+15T+323T2+2665T3+37944T4+2665pT5+323p2T6+15p3T7+p4T8 1 + 15 T + 323 T^{2} + 2665 T^{3} + 37944 T^{4} + 2665 p T^{5} + 323 p^{2} T^{6} + 15 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

−6.96533782031094690438592464026, −6.80242226107146098264162871806, −6.42483563759083864951338429728, −6.39995404412102598077196338387, −6.11416615276011873489365712737, −5.80017717445386214894354939702, −5.59807817860182320034012050669, −5.40075801317743351795784418323, −5.32454458171692868630858501985, −5.24990581075479334580043275570, −4.75895901353658390582609894907, −4.70565326152272854436276361643, −4.53558909667300863919179729368, −4.26364930555437060036039625304, −4.13594955239238313071468468496, −3.74180899562977542287645354573, −3.62498850997454474272934263897, −3.00859459784992687899187139590, −2.81570386255833249466026289327, −2.76802031154388122956489060334, −2.33642005094903179380584020249, −1.85430439698695992713526001106, −1.76686775996363392713797392868, −1.70812417877573680181152914734, −1.18682378364365594590547214346, 0, 0, 0, 0, 1.18682378364365594590547214346, 1.70812417877573680181152914734, 1.76686775996363392713797392868, 1.85430439698695992713526001106, 2.33642005094903179380584020249, 2.76802031154388122956489060334, 2.81570386255833249466026289327, 3.00859459784992687899187139590, 3.62498850997454474272934263897, 3.74180899562977542287645354573, 4.13594955239238313071468468496, 4.26364930555437060036039625304, 4.53558909667300863919179729368, 4.70565326152272854436276361643, 4.75895901353658390582609894907, 5.24990581075479334580043275570, 5.32454458171692868630858501985, 5.40075801317743351795784418323, 5.59807817860182320034012050669, 5.80017717445386214894354939702, 6.11416615276011873489365712737, 6.39995404412102598077196338387, 6.42483563759083864951338429728, 6.80242226107146098264162871806, 6.96533782031094690438592464026

Graph of the ZZ-function along the critical line