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Properties

Label 8-228e4-1.1-c5e4-0-1
Degree $8$
Conductor $2702336256$
Sign $1$
Analytic cond. $1.78805\times 10^{6}$
Root an. cond. $6.04710$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 36·3-s − 84·5-s + 54·7-s + 810·9-s + 354·11-s + 46·13-s + 3.02e3·15-s + 2.04e3·17-s + 1.44e3·19-s − 1.94e3·21-s + 846·23-s − 713·25-s − 1.45e4·27-s − 7.15e3·29-s + 238·31-s − 1.27e4·33-s − 4.53e3·35-s − 1.03e4·37-s − 1.65e3·39-s − 2.82e4·41-s − 1.85e4·43-s − 6.80e4·45-s − 1.86e4·47-s − 4.11e4·49-s − 7.36e4·51-s − 4.27e4·53-s − 2.97e4·55-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.50·5-s + 0.416·7-s + 10/3·9-s + 0.882·11-s + 0.0754·13-s + 3.47·15-s + 1.71·17-s + 0.917·19-s − 0.961·21-s + 0.333·23-s − 0.228·25-s − 3.84·27-s − 1.57·29-s + 0.0444·31-s − 2.03·33-s − 0.625·35-s − 1.24·37-s − 0.174·39-s − 2.61·41-s − 1.53·43-s − 5.00·45-s − 1.23·47-s − 2.44·49-s − 3.96·51-s − 2.08·53-s − 1.32·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.78805\times 10^{6}\)
Root analytic conductor: \(6.04710\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{4} \)
19$C_1$ \( ( 1 - p^{2} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 84 T + 7769 T^{2} + 103914 p T^{3} + 26189784 T^{4} + 103914 p^{6} T^{5} + 7769 p^{10} T^{6} + 84 p^{15} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 54 T + 44045 T^{2} - 3706602 T^{3} + 904289076 T^{4} - 3706602 p^{5} T^{5} + 44045 p^{10} T^{6} - 54 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 354 T + 3877 p^{2} T^{2} - 187554750 T^{3} + 98943513932 T^{4} - 187554750 p^{5} T^{5} + 3877 p^{12} T^{6} - 354 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 46 T + 35120 p T^{2} + 263460534 T^{3} + 39258928526 T^{4} + 263460534 p^{5} T^{5} + 35120 p^{11} T^{6} - 46 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 2046 T + 355169 p T^{2} - 26931522 p^{2} T^{3} + 2599929532 p^{3} T^{4} - 26931522 p^{7} T^{5} + 355169 p^{11} T^{6} - 2046 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 846 T + 14168572 T^{2} - 13344201510 T^{3} + 129565658631590 T^{4} - 13344201510 p^{5} T^{5} + 14168572 p^{10} T^{6} - 846 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 7152 T + 67893836 T^{2} + 256658848848 T^{3} + 1631010975370614 T^{4} + 256658848848 p^{5} T^{5} + 67893836 p^{10} T^{6} + 7152 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 238 T + 60745808 T^{2} + 88058647458 T^{3} + 1883733679822142 T^{4} + 88058647458 p^{5} T^{5} + 60745808 p^{10} T^{6} - 238 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 10326 T + 258607604 T^{2} + 1800193023018 T^{3} + 25877095031647638 T^{4} + 1800193023018 p^{5} T^{5} + 258607604 p^{10} T^{6} + 10326 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 28200 T + 257316208 T^{2} - 488788752840 T^{3} - 25141636924424386 T^{4} - 488788752840 p^{5} T^{5} + 257316208 p^{10} T^{6} + 28200 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 18590 T + 183122693 T^{2} - 1505774882538 T^{3} - 36857819289419860 T^{4} - 1505774882538 p^{5} T^{5} + 183122693 p^{10} T^{6} + 18590 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 18684 T + 743652049 T^{2} + 11253762148026 T^{3} + 234007744957343420 T^{4} + 11253762148026 p^{5} T^{5} + 743652049 p^{10} T^{6} + 18684 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 42720 T + 1231248316 T^{2} + 17780377001952 T^{3} + 342926483045592182 T^{4} + 17780377001952 p^{5} T^{5} + 1231248316 p^{10} T^{6} + 42720 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 68256 T + 4071246940 T^{2} + 152335797210912 T^{3} + 4800425268401564918 T^{4} + 152335797210912 p^{5} T^{5} + 4071246940 p^{10} T^{6} + 68256 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 67654 T + 4178091049 T^{2} + 147966279290878 T^{3} + 5229512626901612236 T^{4} + 147966279290878 p^{5} T^{5} + 4178091049 p^{10} T^{6} + 67654 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 81836 T + 4458869324 T^{2} + 150509647222860 T^{3} + 5721428547728180726 T^{4} + 150509647222860 p^{5} T^{5} + 4458869324 p^{10} T^{6} + 81836 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 62112 T + 5488592620 T^{2} + 316925709623328 T^{3} + 13483742439335474438 T^{4} + 316925709623328 p^{5} T^{5} + 5488592620 p^{10} T^{6} + 62112 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 81966 T + 5312869385 T^{2} + 269490846167082 T^{3} + 14488982878144154412 T^{4} + 269490846167082 p^{5} T^{5} + 5312869385 p^{10} T^{6} + 81966 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 8766 T + 1762510196 T^{2} + 138616649917722 T^{3} - 2405108231355925290 T^{4} + 138616649917722 p^{5} T^{5} + 1762510196 p^{10} T^{6} - 8766 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 55080 T + 10064815516 T^{2} + 775985455694952 T^{3} + 47429489921323557782 T^{4} + 775985455694952 p^{5} T^{5} + 10064815516 p^{10} T^{6} + 55080 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 197520 T + 26785146172 T^{2} + 2826246029891568 T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + 2826246029891568 p^{5} T^{5} + 26785146172 p^{10} T^{6} + 197520 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 184228 T + 27810004324 T^{2} + 2981910942267100 T^{3} + \)\(33\!\cdots\!06\)\( T^{4} + 2981910942267100 p^{5} T^{5} + 27810004324 p^{10} T^{6} + 184228 p^{15} T^{7} + p^{20} T^{8} \)
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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

−8.420791639059694163906969844323, −7.951027536346728710677065593209, −7.84324441103473051184594112249, −7.74947300536159163712199067726, −7.65842540762743278037804324434, −7.08849057478199518212121584476, −6.72622463387789191865043925324, −6.65576838940609125180053940122, −6.63286222265892512542610253891, −5.89072503094396761514538369819, −5.77933115131253201887622527326, −5.52385528523660650503167087986, −5.39892691313244505144072908636, −4.89658093495591602661315327371, −4.50729648778957691968605437370, −4.48251429706451279493444018302, −4.34468583787043453428322464328, −3.47308407854136338276212550315, −3.40702552182980058682690931390, −3.30542582375173871886603907694, −2.96766717665306792285172992481, −1.67477364996538124978858523759, −1.47623327758036045456946391414, −1.43965225288714187176348498036, −1.36009506332972144178562403118, 0, 0, 0, 0, 1.36009506332972144178562403118, 1.43965225288714187176348498036, 1.47623327758036045456946391414, 1.67477364996538124978858523759, 2.96766717665306792285172992481, 3.30542582375173871886603907694, 3.40702552182980058682690931390, 3.47308407854136338276212550315, 4.34468583787043453428322464328, 4.48251429706451279493444018302, 4.50729648778957691968605437370, 4.89658093495591602661315327371, 5.39892691313244505144072908636, 5.52385528523660650503167087986, 5.77933115131253201887622527326, 5.89072503094396761514538369819, 6.63286222265892512542610253891, 6.65576838940609125180053940122, 6.72622463387789191865043925324, 7.08849057478199518212121584476, 7.65842540762743278037804324434, 7.74947300536159163712199067726, 7.84324441103473051184594112249, 7.951027536346728710677065593209, 8.420791639059694163906969844323

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