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Properties

Label 8-1224e4-1.1-c3e4-0-1
Degree $8$
Conductor $2.245\times 10^{12}$
Sign $1$
Analytic cond. $2.72013\times 10^{7}$
Root an. cond. $8.49813$
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  − 8·5-s − 22·7-s − 70·11-s + 120·13-s + 68·17-s − 44·19-s − 158·23-s + 56·25-s − 264·29-s + 122·31-s + 176·35-s + 256·37-s − 240·41-s − 1.10e3·43-s + 800·47-s − 438·49-s − 432·53-s + 560·55-s + 148·59-s − 728·61-s − 960·65-s − 1.03e3·67-s − 798·71-s − 1.54e3·73-s + 1.54e3·77-s + 758·79-s − 244·83-s + ⋯
L(s)  = 1  − 0.715·5-s − 1.18·7-s − 1.91·11-s + 2.56·13-s + 0.970·17-s − 0.531·19-s − 1.43·23-s + 0.447·25-s − 1.69·29-s + 0.706·31-s + 0.849·35-s + 1.13·37-s − 0.914·41-s − 3.90·43-s + 2.48·47-s − 1.27·49-s − 1.11·53-s + 1.37·55-s + 0.326·59-s − 1.52·61-s − 1.83·65-s − 1.88·67-s − 1.33·71-s − 2.47·73-s + 2.27·77-s + 1.07·79-s − 0.322·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(2.72013\times 10^{7}\)
Root analytic conductor: \(8.49813\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 17^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
3 \( 1 \)
17$C_1$ \( ( 1 - p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 8 T + 8 T^{2} + 472 T^{3} + 20414 T^{4} + 472 p^{3} T^{5} + 8 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 22 T + 922 T^{2} + 15278 T^{3} + 450602 T^{4} + 15278 p^{3} T^{5} + 922 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 70 T + 5294 T^{2} + 261054 T^{3} + 10506858 T^{4} + 261054 p^{3} T^{5} + 5294 p^{6} T^{6} + 70 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 120 T + 9184 T^{2} - 509624 T^{3} + 26258478 T^{4} - 509624 p^{3} T^{5} + 9184 p^{6} T^{6} - 120 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 44 T + 2836 T^{2} + 24492 T^{3} + 46895430 T^{4} + 24492 p^{3} T^{5} + 2836 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 158 T + 7730 T^{2} - 365034 T^{3} - 120322470 T^{4} - 365034 p^{3} T^{5} + 7730 p^{6} T^{6} + 158 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 264 T + 2056 p T^{2} + 8835992 T^{3} + 1500327390 T^{4} + 8835992 p^{3} T^{5} + 2056 p^{7} T^{6} + 264 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 122 T + 80002 T^{2} - 159470 p T^{3} + 2814850330 T^{4} - 159470 p^{4} T^{5} + 80002 p^{6} T^{6} - 122 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 256 T + 176248 T^{2} - 33248832 T^{3} + 12972545886 T^{4} - 33248832 p^{3} T^{5} + 176248 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 240 T + 199292 T^{2} + 34097808 T^{3} + 18346209126 T^{4} + 34097808 p^{3} T^{5} + 199292 p^{6} T^{6} + 240 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 1100 T + 735364 T^{2} + 328732844 T^{3} + 108061258438 T^{4} + 328732844 p^{3} T^{5} + 735364 p^{6} T^{6} + 1100 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 800 T + 541196 T^{2} - 218719392 T^{3} + 83917026726 T^{4} - 218719392 p^{3} T^{5} + 541196 p^{6} T^{6} - 800 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 432 T + 551708 T^{2} + 176382800 T^{3} + 119184343638 T^{4} + 176382800 p^{3} T^{5} + 551708 p^{6} T^{6} + 432 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 148 T + 421892 T^{2} - 75180020 T^{3} + 117948373766 T^{4} - 75180020 p^{3} T^{5} + 421892 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 728 T + 833704 T^{2} + 391040872 T^{3} + 264152111774 T^{4} + 391040872 p^{3} T^{5} + 833704 p^{6} T^{6} + 728 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1032 T + 1153276 T^{2} + 758126600 T^{3} + 495040471158 T^{4} + 758126600 p^{3} T^{5} + 1153276 p^{6} T^{6} + 1032 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 798 T + 1106522 T^{2} + 704044214 T^{3} + 7614984870 p T^{4} + 704044214 p^{3} T^{5} + 1106522 p^{6} T^{6} + 798 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1544 T + 2218780 T^{2} + 1834331384 T^{3} + 1409790021670 T^{4} + 1834331384 p^{3} T^{5} + 2218780 p^{6} T^{6} + 1544 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 758 T + 1859842 T^{2} - 1100794910 T^{3} + 1350325742842 T^{4} - 1100794910 p^{3} T^{5} + 1859842 p^{6} T^{6} - 758 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 244 T + 1969316 T^{2} + 307278644 T^{3} + 1592927896358 T^{4} + 307278644 p^{3} T^{5} + 1969316 p^{6} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1440 T + 2610320 T^{2} + 2559236336 T^{3} + 2640214825182 T^{4} + 2559236336 p^{3} T^{5} + 2610320 p^{6} T^{6} + 1440 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1344 T + 1760716 T^{2} - 95293952 T^{3} + 5534915430 T^{4} - 95293952 p^{3} T^{5} + 1760716 p^{6} T^{6} + 1344 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.98688643911957984370546921601, −6.65976677612443920294524505052, −6.56840549303039127445638321819, −6.25011746902165221672339357438, −6.09042081282977853020708120959, −5.89545714557619521156146917346, −5.73687398727511491125948791547, −5.61291809872383899020800030637, −5.35336259961394527096888000711, −4.74325991743927202729189002679, −4.72801985666802285361774760024, −4.59225336338634544347124448485, −4.38781195327918102960706258672, −3.69380509725596781655497567488, −3.69033036363547139252019528417, −3.50633646162999313600774502550, −3.48127318060101320750173631828, −3.08664695622429345719519085069, −2.76172187437408384995166939505, −2.51796281006299699205435293961, −2.30791320813636322743701126522, −1.64623528232425885852377977101, −1.55582683836812055002212819627, −1.21587830942174538111291604010, −1.05150977474552458260882841636, 0, 0, 0, 0, 1.05150977474552458260882841636, 1.21587830942174538111291604010, 1.55582683836812055002212819627, 1.64623528232425885852377977101, 2.30791320813636322743701126522, 2.51796281006299699205435293961, 2.76172187437408384995166939505, 3.08664695622429345719519085069, 3.48127318060101320750173631828, 3.50633646162999313600774502550, 3.69033036363547139252019528417, 3.69380509725596781655497567488, 4.38781195327918102960706258672, 4.59225336338634544347124448485, 4.72801985666802285361774760024, 4.74325991743927202729189002679, 5.35336259961394527096888000711, 5.61291809872383899020800030637, 5.73687398727511491125948791547, 5.89545714557619521156146917346, 6.09042081282977853020708120959, 6.25011746902165221672339357438, 6.56840549303039127445638321819, 6.65976677612443920294524505052, 6.98688643911957984370546921601

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