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Properties

Label 8-1232e4-1.1-c3e4-0-2
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $2.79194\times 10^{7}$
Root an. cond. $8.52586$
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  + 3·3-s + 5-s + 28·7-s − 59·9-s − 44·11-s + 98·13-s + 3·15-s + 64·17-s − 114·19-s + 84·21-s − 231·23-s − 41·25-s − 234·27-s + 268·29-s − 33·31-s − 132·33-s + 28·35-s + 357·37-s + 294·39-s + 364·41-s − 44·43-s − 59·45-s − 720·47-s + 490·49-s + 192·51-s + 740·53-s − 44·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.0894·5-s + 1.51·7-s − 2.18·9-s − 1.20·11-s + 2.09·13-s + 0.0516·15-s + 0.913·17-s − 1.37·19-s + 0.872·21-s − 2.09·23-s − 0.327·25-s − 1.66·27-s + 1.71·29-s − 0.191·31-s − 0.696·33-s + 0.135·35-s + 1.58·37-s + 1.20·39-s + 1.38·41-s − 0.156·43-s − 0.195·45-s − 2.23·47-s + 10/7·49-s + 0.527·51-s + 1.91·53-s − 0.107·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.79194\times 10^{7}\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.675991778\)
\(L(\frac12)\) \(\approx\) \(6.675991778\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
11$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - p T + 68 T^{2} - 49 p T^{3} + 794 p T^{4} - 49 p^{4} T^{5} + 68 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - T + 42 T^{2} + 861 T^{3} + 11434 T^{4} + 861 p^{3} T^{5} + 42 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 98 T + 8160 T^{2} - 379102 T^{3} + 20486158 T^{4} - 379102 p^{3} T^{5} + 8160 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 64 T + 1484 T^{2} - 232712 T^{3} + 47199350 T^{4} - 232712 p^{3} T^{5} + 1484 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6 p T + 25036 T^{2} + 2325906 T^{3} + 249538710 T^{4} + 2325906 p^{3} T^{5} + 25036 p^{6} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 231 T + 54852 T^{2} + 7672947 T^{3} + 1045005974 T^{4} + 7672947 p^{3} T^{5} + 54852 p^{6} T^{6} + 231 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 268 T + 101716 T^{2} - 18029252 T^{3} + 3709626758 T^{4} - 18029252 p^{3} T^{5} + 101716 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 33 T + 54484 T^{2} + 223333 T^{3} + 2139658734 T^{4} + 223333 p^{3} T^{5} + 54484 p^{6} T^{6} + 33 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 357 T + 127390 T^{2} - 33734767 T^{3} + 9968259426 T^{4} - 33734767 p^{3} T^{5} + 127390 p^{6} T^{6} - 357 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 364 T + 159364 T^{2} - 13290332 T^{3} + 6070138310 T^{4} - 13290332 p^{3} T^{5} + 159364 p^{6} T^{6} - 364 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 44 T + 261676 T^{2} + 2944972 T^{3} + 28745483702 T^{4} + 2944972 p^{3} T^{5} + 261676 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 720 T + 416684 T^{2} + 164392952 T^{3} + 61032189222 T^{4} + 164392952 p^{3} T^{5} + 416684 p^{6} T^{6} + 720 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 740 T + 715364 T^{2} - 319275756 T^{3} + 166686388614 T^{4} - 319275756 p^{3} T^{5} + 715364 p^{6} T^{6} - 740 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 787 T + 780236 T^{2} + 414558699 T^{3} + 240794268750 T^{4} + 414558699 p^{3} T^{5} + 780236 p^{6} T^{6} + 787 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1020 T + 995596 T^{2} - 635641948 T^{3} + 350297020038 T^{4} - 635641948 p^{3} T^{5} + 995596 p^{6} T^{6} - 1020 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 995 T + 1065272 T^{2} - 808094883 T^{3} + 461986441118 T^{4} - 808094883 p^{3} T^{5} + 1065272 p^{6} T^{6} - 995 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1011 T + 1109060 T^{2} - 483905855 T^{3} + 381085044822 T^{4} - 483905855 p^{3} T^{5} + 1109060 p^{6} T^{6} - 1011 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1592 T + 2328428 T^{2} - 1950945600 T^{3} + 1500784205942 T^{4} - 1950945600 p^{3} T^{5} + 2328428 p^{6} T^{6} - 1592 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 178 T + 1447084 T^{2} - 53555370 T^{3} + 925890126822 T^{4} - 53555370 p^{3} T^{5} + 1447084 p^{6} T^{6} - 178 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 324 T + 285052 T^{2} - 30403452 T^{3} - 164201388106 T^{4} - 30403452 p^{3} T^{5} + 285052 p^{6} T^{6} + 324 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 19 T + 1836758 T^{2} - 502734021 T^{3} + 1537607385138 T^{4} - 502734021 p^{3} T^{5} + 1836758 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 555 T + 1641394 T^{2} + 2039307557 T^{3} + 1335907909242 T^{4} + 2039307557 p^{3} T^{5} + 1641394 p^{6} T^{6} + 555 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.55078937621559271922559234614, −6.17269375604568857038607724592, −6.07544766362824204217127173253, −5.84672931427160532408678771216, −5.75974940957905866312278343162, −5.50639898964938936734927263108, −5.09217084864282387255551588473, −5.07140146670454333843795028229, −4.96444230619744571358898099553, −4.39981572452100776390832556960, −4.07342183004301640470292852310, −3.99437143584560705631091614903, −3.98190354729542565337931898118, −3.46697539925001940688716896771, −3.10495978109020133994994490383, −3.06161202067350457889291945722, −2.74376816548591586344133140064, −2.26334795758552018803584095807, −2.22780415439556682985418673262, −1.98452800546254252892634168356, −1.76735724971766682439455203020, −1.15431984900045081637534242259, −0.76264664833339670601598011321, −0.69876321166755813704292746145, −0.29122667016107637173099446462, 0.29122667016107637173099446462, 0.69876321166755813704292746145, 0.76264664833339670601598011321, 1.15431984900045081637534242259, 1.76735724971766682439455203020, 1.98452800546254252892634168356, 2.22780415439556682985418673262, 2.26334795758552018803584095807, 2.74376816548591586344133140064, 3.06161202067350457889291945722, 3.10495978109020133994994490383, 3.46697539925001940688716896771, 3.98190354729542565337931898118, 3.99437143584560705631091614903, 4.07342183004301640470292852310, 4.39981572452100776390832556960, 4.96444230619744571358898099553, 5.07140146670454333843795028229, 5.09217084864282387255551588473, 5.50639898964938936734927263108, 5.75974940957905866312278343162, 5.84672931427160532408678771216, 6.07544766362824204217127173253, 6.17269375604568857038607724592, 6.55078937621559271922559234614

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