Properties

Label 8-2352e4-1.1-c3e4-0-0
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $3.70863\times 10^{8}$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s + 4·5-s + 90·9-s − 14·11-s + 22·13-s − 48·15-s + 96·17-s + 26·19-s − 96·23-s − 187·25-s − 540·27-s − 76·29-s − 238·31-s + 168·33-s + 562·37-s − 264·39-s + 428·41-s + 258·43-s + 360·45-s + 80·47-s − 1.15e3·51-s − 56·55-s − 312·57-s − 262·59-s − 276·61-s + 88·65-s − 150·67-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.357·5-s + 10/3·9-s − 0.383·11-s + 0.469·13-s − 0.826·15-s + 1.36·17-s + 0.313·19-s − 0.870·23-s − 1.49·25-s − 3.84·27-s − 0.486·29-s − 1.37·31-s + 0.886·33-s + 2.49·37-s − 1.08·39-s + 1.63·41-s + 0.914·43-s + 1.19·45-s + 0.248·47-s − 3.16·51-s − 0.137·55-s − 0.725·57-s − 0.578·59-s − 0.579·61-s + 0.167·65-s − 0.273·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.70863\times 10^{8}\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.04484609298\)
\(L(\frac12)\) \(\approx\) \(0.04484609298\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
7 \( 1 \)
good5$C_2 \wr S_4$ \( 1 - 4 T + 203 T^{2} - 1244 T^{3} + 19472 T^{4} - 1244 p^{3} T^{5} + 203 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 14 T - 71 T^{2} - 12014 T^{3} + 2033720 T^{4} - 12014 p^{3} T^{5} - 71 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 22 T + 105 p T^{2} - 60998 T^{3} + 10139200 T^{4} - 60998 p^{3} T^{5} + 105 p^{7} T^{6} - 22 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 96 T + 5348 T^{2} + 301088 T^{3} - 39928698 T^{4} + 301088 p^{3} T^{5} + 5348 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 26 T + 13201 T^{2} - 793654 T^{3} + 92486852 T^{4} - 793654 p^{3} T^{5} + 13201 p^{6} T^{6} - 26 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 96 T + 34364 T^{2} + 3625760 T^{3} + 540043494 T^{4} + 3625760 p^{3} T^{5} + 34364 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 76 T + 45263 T^{2} + 1864308 T^{3} + 1017389172 T^{4} + 1864308 p^{3} T^{5} + 45263 p^{6} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 238 T + 40396 T^{2} + 4512200 T^{3} + 466839125 T^{4} + 4512200 p^{3} T^{5} + 40396 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 562 T + 218917 T^{2} - 51298938 T^{3} + 12421701936 T^{4} - 51298938 p^{3} T^{5} + 218917 p^{6} T^{6} - 562 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 428 T + 234552 T^{2} - 51690756 T^{3} + 18963959182 T^{4} - 51690756 p^{3} T^{5} + 234552 p^{6} T^{6} - 428 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 6 p T + 138485 T^{2} - 14865486 T^{3} + 6723608208 T^{4} - 14865486 p^{3} T^{5} + 138485 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 80 T + 79044 T^{2} - 6436176 T^{3} + 18721328134 T^{4} - 6436176 p^{3} T^{5} + 79044 p^{6} T^{6} - 80 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 8683 p T^{2} + 3013660 T^{3} + 92692295988 T^{4} + 3013660 p^{3} T^{5} + 8683 p^{7} T^{6} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 262 T + 544965 T^{2} + 206100090 T^{3} + 137845655068 T^{4} + 206100090 p^{3} T^{5} + 544965 p^{6} T^{6} + 262 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 276 T + 472228 T^{2} + 230097500 T^{3} + 121196455014 T^{4} + 230097500 p^{3} T^{5} + 472228 p^{6} T^{6} + 276 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 150 T + 612301 T^{2} + 160051070 T^{3} + 213928110456 T^{4} + 160051070 p^{3} T^{5} + 612301 p^{6} T^{6} + 150 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 848 T + 1353124 T^{2} - 864361552 T^{3} + 714336102950 T^{4} - 864361552 p^{3} T^{5} + 1353124 p^{6} T^{6} - 848 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 218 T + 1524553 T^{2} + 248776730 T^{3} + 12106383700 p T^{4} + 248776730 p^{3} T^{5} + 1524553 p^{6} T^{6} + 218 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1762 T + 2715892 T^{2} + 2479590568 T^{3} + 2098857820333 T^{4} + 2479590568 p^{3} T^{5} + 2715892 p^{6} T^{6} + 1762 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 3450 T + 6524297 T^{2} + 8058006346 T^{3} + 7159809209520 T^{4} + 8058006346 p^{3} T^{5} + 6524297 p^{6} T^{6} + 3450 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 344 T + 1670240 T^{2} + 134058408 T^{3} + 1315175698974 T^{4} + 134058408 p^{3} T^{5} + 1670240 p^{6} T^{6} + 344 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 622 T + 849993 T^{2} - 199355338 T^{3} - 35005181276 T^{4} - 199355338 p^{3} T^{5} + 849993 p^{6} T^{6} + 622 p^{9} T^{7} + p^{12} T^{8} \)
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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

−5.95223946637077819540945921463, −5.85506932340954660010160137992, −5.65666034388256730761791038784, −5.58642184740769242236728742741, −5.27504095196416898358072808758, −5.12318318368303619776479128412, −4.75673644567084260827829007025, −4.55784687937678400751593365102, −4.39206635207382981652551897374, −4.09091788711635807199624757278, −4.03849817760372907622874407004, −3.72214078368920100702880548772, −3.67924591047356129973775489867, −3.12026182343268690991216948924, −2.88358389202448346141835588957, −2.69293480961935627668399063972, −2.47355632328752584945995511849, −2.04616295810099292924433657715, −1.68835392632215501655019969494, −1.53484291114964680610248292946, −1.41508073595588284974086924793, −0.878803159018816621674037834486, −0.817718502839497621070023924337, −0.45181444339146992058102315181, −0.03379922471162098160819472094, 0.03379922471162098160819472094, 0.45181444339146992058102315181, 0.817718502839497621070023924337, 0.878803159018816621674037834486, 1.41508073595588284974086924793, 1.53484291114964680610248292946, 1.68835392632215501655019969494, 2.04616295810099292924433657715, 2.47355632328752584945995511849, 2.69293480961935627668399063972, 2.88358389202448346141835588957, 3.12026182343268690991216948924, 3.67924591047356129973775489867, 3.72214078368920100702880548772, 4.03849817760372907622874407004, 4.09091788711635807199624757278, 4.39206635207382981652551897374, 4.55784687937678400751593365102, 4.75673644567084260827829007025, 5.12318318368303619776479128412, 5.27504095196416898358072808758, 5.58642184740769242236728742741, 5.65666034388256730761791038784, 5.85506932340954660010160137992, 5.95223946637077819540945921463

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