Properties

Label 8-29e4-1.1-c5e4-0-0
Degree $8$
Conductor $707281$
Sign $1$
Analytic cond. $467.987$
Root an. cond. $2.15664$
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  − 28·3-s − 59·4-s − 68·5-s − 208·7-s − 168·8-s − 234·9-s − 124·11-s + 1.65e3·12-s − 460·13-s + 1.90e3·15-s + 1.55e3·16-s + 184·17-s − 2.39e3·19-s + 4.01e3·20-s + 5.82e3·21-s − 1.19e3·23-s + 4.70e3·24-s − 3.02e3·25-s + 1.69e4·27-s + 1.22e4·28-s − 3.36e3·29-s − 1.92e4·31-s + 1.44e4·32-s + 3.47e3·33-s + 1.41e4·35-s + 1.38e4·36-s − 1.09e4·37-s + ⋯
L(s)  = 1  − 1.79·3-s − 1.84·4-s − 1.21·5-s − 1.60·7-s − 0.928·8-s − 0.962·9-s − 0.308·11-s + 3.31·12-s − 0.754·13-s + 2.18·15-s + 1.52·16-s + 0.154·17-s − 1.52·19-s + 2.24·20-s + 2.88·21-s − 0.469·23-s + 1.66·24-s − 0.968·25-s + 4.47·27-s + 2.95·28-s − 0.742·29-s − 3.59·31-s + 2.49·32-s + 0.555·33-s + 1.95·35-s + 1.77·36-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 707281 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 707281 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(707281\)    =    \(29^{4}\)
Sign: $1$
Analytic conductor: \(467.987\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{29} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 707281,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)\)

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)$
bad29$C_1$ \( ( 1 + p^{2} T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 59 T^{2} + 21 p^{3} T^{3} + 481 p^{2} T^{4} + 21 p^{8} T^{5} + 59 p^{10} T^{6} + p^{20} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 28 T + 1018 T^{2} + 6032 p T^{3} + 42035 p^{2} T^{4} + 6032 p^{6} T^{5} + 1018 p^{10} T^{6} + 28 p^{15} T^{7} + p^{20} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 68 T + 306 p^{2} T^{2} + 507792 T^{3} + 34175739 T^{4} + 507792 p^{5} T^{5} + 306 p^{12} T^{6} + 68 p^{15} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 208 T + 50020 T^{2} + 6850192 T^{3} + 1014497510 T^{4} + 6850192 p^{5} T^{5} + 50020 p^{10} T^{6} + 208 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 124 T + 424234 T^{2} - 2589040 T^{3} + 81054467547 T^{4} - 2589040 p^{5} T^{5} + 424234 p^{10} T^{6} + 124 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 460 T + 659346 T^{2} - 118850704 T^{3} + 97850261579 T^{4} - 118850704 p^{5} T^{5} + 659346 p^{10} T^{6} + 460 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 184 T + 5453956 T^{2} - 769013352 T^{3} + 11467152125510 T^{4} - 769013352 p^{5} T^{5} + 5453956 p^{10} T^{6} - 184 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2392 T + 6973884 T^{2} + 12798819480 T^{3} + 24894016673110 T^{4} + 12798819480 p^{5} T^{5} + 6973884 p^{10} T^{6} + 2392 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 1192 T + 18381660 T^{2} + 25053529224 T^{3} + 155801395273446 T^{4} + 25053529224 p^{5} T^{5} + 18381660 p^{10} T^{6} + 1192 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 19212 T + 221298386 T^{2} + 1734150261552 T^{3} + 10610786009786723 T^{4} + 1734150261552 p^{5} T^{5} + 221298386 p^{10} T^{6} + 19212 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 10928 T + 259081044 T^{2} + 2170552279376 T^{3} + 26330320398327638 T^{4} + 2170552279376 p^{5} T^{5} + 259081044 p^{10} T^{6} + 10928 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 1120 T + 124939436 T^{2} + 125715602080 T^{3} + 3124670729197126 T^{4} + 125715602080 p^{5} T^{5} + 124939436 p^{10} T^{6} + 1120 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 21420 T + 435014346 T^{2} + 7359136793872 T^{3} + 89454145102817595 T^{4} + 7359136793872 p^{5} T^{5} + 435014346 p^{10} T^{6} + 21420 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 23772 T + 805388370 T^{2} - 10338335631888 T^{3} + 228527211259419395 T^{4} - 10338335631888 p^{5} T^{5} + 805388370 p^{10} T^{6} - 23772 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 8860 T + 21492794 p T^{2} - 7982732306416 T^{3} + 619291319107778907 T^{4} - 7982732306416 p^{5} T^{5} + 21492794 p^{11} T^{6} - 8860 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 10840 T + 1701897548 T^{2} + 4343203272088 T^{3} + 1367718152106467222 T^{4} + 4343203272088 p^{5} T^{5} + 1701897548 p^{10} T^{6} + 10840 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 49448 T + 2057321812 T^{2} - 78522043066328 T^{3} + 2952566139994453046 T^{4} - 78522043066328 p^{5} T^{5} + 2057321812 p^{10} T^{6} - 49448 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 7840 T + 2717901676 T^{2} + 29569800257440 T^{3} + 3898941673833682358 T^{4} + 29569800257440 p^{5} T^{5} + 2717901676 p^{10} T^{6} + 7840 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 48744 T + 6548351764 T^{2} + 230106406041576 T^{3} + 16930032452217232710 T^{4} + 230106406041576 p^{5} T^{5} + 6548351764 p^{10} T^{6} + 48744 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 74992 T + 7968026628 T^{2} + 382291369814608 T^{3} + 24038082722458524710 T^{4} + 382291369814608 p^{5} T^{5} + 7968026628 p^{10} T^{6} + 74992 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 106076 T + 11710838338 T^{2} + 694262350746992 T^{3} + 47791838682726576947 T^{4} + 694262350746992 p^{5} T^{5} + 11710838338 p^{10} T^{6} + 106076 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 62888 T + 14415496844 T^{2} - 604874915361704 T^{3} + 80649041467245964598 T^{4} - 604874915361704 p^{5} T^{5} + 14415496844 p^{10} T^{6} - 62888 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 107568 T + 18707891628 T^{2} - 1720808168131152 T^{3} + \)\(14\!\cdots\!82\)\( T^{4} - 1720808168131152 p^{5} T^{5} + 18707891628 p^{10} T^{6} - 107568 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 49520 T + 17287734156 T^{2} + 2140548822317840 T^{3} + \)\(13\!\cdots\!58\)\( T^{4} + 2140548822317840 p^{5} T^{5} + 17287734156 p^{10} T^{6} + 49520 p^{15} T^{7} + p^{20} T^{8} \)
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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

−12.51840770723615562574405252373, −11.95045393567753199675588338820, −11.80070269425034457392820771438, −11.72950213981280341246007594859, −11.12418386082560822681824002023, −11.07751438420669907560365211628, −10.37312581913369923034260356122, −10.01668248323960477023189601115, −9.944047403584147838425541924254, −9.072611988000810662112591225342, −9.006756722137752417052934671566, −8.700123517443802261234511953102, −8.549307801629179520181945231379, −7.62839149562236999932527429138, −7.60939721450575787916035731935, −6.79019200518989928730526496808, −6.28412900245602665615722872437, −5.95550225033633985429419948873, −5.70081226594538631404387762560, −5.16879042930089851983957118009, −4.97366964083269778478972425768, −4.06582300883637220147848087199, −3.63547011652282956089496635579, −3.33815165503288978586231114877, −2.38617031077110863424813557446, 0, 0, 0, 0, 2.38617031077110863424813557446, 3.33815165503288978586231114877, 3.63547011652282956089496635579, 4.06582300883637220147848087199, 4.97366964083269778478972425768, 5.16879042930089851983957118009, 5.70081226594538631404387762560, 5.95550225033633985429419948873, 6.28412900245602665615722872437, 6.79019200518989928730526496808, 7.60939721450575787916035731935, 7.62839149562236999932527429138, 8.549307801629179520181945231379, 8.700123517443802261234511953102, 9.006756722137752417052934671566, 9.072611988000810662112591225342, 9.944047403584147838425541924254, 10.01668248323960477023189601115, 10.37312581913369923034260356122, 11.07751438420669907560365211628, 11.12418386082560822681824002023, 11.72950213981280341246007594859, 11.80070269425034457392820771438, 11.95045393567753199675588338820, 12.51840770723615562574405252373

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