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Properties

Label 8-1792e4-1.1-c3e4-0-6
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $1.24973\times 10^{8}$
Root an. cond. $10.2825$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 10·5-s − 28·7-s − 34·9-s + 64·11-s − 14·13-s − 60·15-s + 40·17-s + 242·19-s + 168·21-s − 64·23-s − 286·25-s + 330·27-s − 524·29-s − 24·31-s − 384·33-s − 280·35-s + 212·37-s + 84·39-s + 16·41-s + 128·43-s − 340·45-s − 296·47-s + 490·49-s − 240·51-s + 408·53-s + 640·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1.51·7-s − 1.25·9-s + 1.75·11-s − 0.298·13-s − 1.03·15-s + 0.570·17-s + 2.92·19-s + 1.74·21-s − 0.580·23-s − 2.28·25-s + 2.35·27-s − 3.35·29-s − 0.139·31-s − 2.02·33-s − 1.35·35-s + 0.941·37-s + 0.344·39-s + 0.0609·41-s + 0.453·43-s − 1.12·45-s − 0.918·47-s + 10/7·49-s − 0.658·51-s + 1.05·53-s + 1.56·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.24973\times 10^{8}\)
Root analytic conductor: \(10.2825\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T + 70 T^{2} + 98 p T^{3} + 2506 T^{4} + 98 p^{4} T^{5} + 70 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 2 p T + 386 T^{2} - 2902 T^{3} + 67522 T^{4} - 2902 p^{3} T^{5} + 386 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 64 T + 3896 T^{2} - 15824 p T^{3} + 7493854 T^{4} - 15824 p^{4} T^{5} + 3896 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 5498 T^{2} + 86562 T^{3} + 16581362 T^{4} + 86562 p^{3} T^{5} + 5498 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 732 p T^{2} - 634008 T^{3} + 77656582 T^{4} - 634008 p^{3} T^{5} + 732 p^{7} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 242 T + 39470 T^{2} - 4520082 T^{3} + 417177242 T^{4} - 4520082 p^{3} T^{5} + 39470 p^{6} T^{6} - 242 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 64 T + 42000 T^{2} + 2010432 T^{3} + 723262078 T^{4} + 2010432 p^{3} T^{5} + 42000 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 524 T + 6204 p T^{2} + 42453300 T^{3} + 7621276870 T^{4} + 42453300 p^{3} T^{5} + 6204 p^{7} T^{6} + 524 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 36044 T^{2} + 4119480 T^{3} + 1278352614 T^{4} + 4119480 p^{3} T^{5} + 36044 p^{6} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 212 T + 150316 T^{2} - 31713164 T^{3} + 10354140262 T^{4} - 31713164 p^{3} T^{5} + 150316 p^{6} T^{6} - 212 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 105276 T^{2} + 26099856 T^{3} + 4672195558 T^{4} + 26099856 p^{3} T^{5} + 105276 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 128 T + 164408 T^{2} - 16655664 T^{3} + 14272413278 T^{4} - 16655664 p^{3} T^{5} + 164408 p^{6} T^{6} - 128 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 296 T + 361692 T^{2} + 73372680 T^{3} + 52799197702 T^{4} + 73372680 p^{3} T^{5} + 361692 p^{6} T^{6} + 296 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 408 T + 401156 T^{2} - 145430568 T^{3} + 86189893398 T^{4} - 145430568 p^{3} T^{5} + 401156 p^{6} T^{6} - 408 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 355950 T^{2} + 68243550 T^{3} + 88423182170 T^{4} + 68243550 p^{3} T^{5} + 355950 p^{6} T^{6} + 30 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 1542 T + 1774514 T^{2} + 1260146922 T^{3} + 721004309154 T^{4} + 1260146922 p^{3} T^{5} + 1774514 p^{6} T^{6} + 1542 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 116 T + 583792 T^{2} - 249186932 T^{3} + 161538965998 T^{4} - 249186932 p^{3} T^{5} + 583792 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 584 T + 542460 T^{2} + 14444184 T^{3} + 67443285286 T^{4} + 14444184 p^{3} T^{5} + 542460 p^{6} T^{6} - 584 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 768 T + 947308 T^{2} - 808867008 T^{3} + 437175361158 T^{4} - 808867008 p^{3} T^{5} + 947308 p^{6} T^{6} - 768 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 776 T + 1891676 T^{2} + 1136400744 T^{3} + 1380231878534 T^{4} + 1136400744 p^{3} T^{5} + 1891676 p^{6} T^{6} + 776 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 346 T + 1652694 T^{2} + 334091466 T^{3} + 1248337154794 T^{4} + 334091466 p^{3} T^{5} + 1652694 p^{6} T^{6} + 346 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 2576 T + 4515324 T^{2} + 5400247152 T^{3} + 5161182144166 T^{4} + 5400247152 p^{3} T^{5} + 4515324 p^{6} T^{6} + 2576 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 1640 T + 39340 p T^{2} - 4338255896 T^{3} + 5279068197670 T^{4} - 4338255896 p^{3} T^{5} + 39340 p^{7} T^{6} - 1640 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.58166891430770632787677017358, −6.14442012062625121664975790812, −6.10587177700728025926072418152, −6.10120650041089170544797196311, −5.86559925542076908189683930634, −5.54186136034161295973828079268, −5.48218424092110337320383507786, −5.39082608780340460260801003466, −5.27358825480686978743138606561, −4.67508227282946156147148225507, −4.59743556808234563688900087822, −4.22682411143487448359598599554, −3.96518121927911195980889541022, −3.57909308567173485588196778732, −3.45437366076165454956388074306, −3.44833468288177920982009429509, −3.37865697007248382818063437850, −2.67930841841126385846830503261, −2.51059801258180618609451634447, −2.35820351379447903842759054751, −2.13627120382050687296604916151, −1.43095542657994669799759275770, −1.37386777760233877955749447391, −1.21116451652393028026630491186, −0.929051388972251069381563186020, 0, 0, 0, 0, 0.929051388972251069381563186020, 1.21116451652393028026630491186, 1.37386777760233877955749447391, 1.43095542657994669799759275770, 2.13627120382050687296604916151, 2.35820351379447903842759054751, 2.51059801258180618609451634447, 2.67930841841126385846830503261, 3.37865697007248382818063437850, 3.44833468288177920982009429509, 3.45437366076165454956388074306, 3.57909308567173485588196778732, 3.96518121927911195980889541022, 4.22682411143487448359598599554, 4.59743556808234563688900087822, 4.67508227282946156147148225507, 5.27358825480686978743138606561, 5.39082608780340460260801003466, 5.48218424092110337320383507786, 5.54186136034161295973828079268, 5.86559925542076908189683930634, 6.10120650041089170544797196311, 6.10587177700728025926072418152, 6.14442012062625121664975790812, 6.58166891430770632787677017358

Graph of the $Z$-function along the critical line