Properties

Label 8-230e4-1.1-c3e4-0-0
Degree $8$
Conductor $2798410000$
Sign $1$
Analytic cond. $33913.7$
Root an. cond. $3.68380$
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  − 8·2-s − 4·3-s + 40·4-s − 20·5-s + 32·6-s − 7-s − 160·8-s − 30·9-s + 160·10-s − 39·11-s − 160·12-s − 20·13-s + 8·14-s + 80·15-s + 560·16-s − 23·17-s + 240·18-s + 53·19-s − 800·20-s + 4·21-s + 312·22-s − 92·23-s + 640·24-s + 250·25-s + 160·26-s + 223·27-s − 40·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.769·3-s + 5·4-s − 1.78·5-s + 2.17·6-s − 0.0539·7-s − 7.07·8-s − 1.11·9-s + 5.05·10-s − 1.06·11-s − 3.84·12-s − 0.426·13-s + 0.152·14-s + 1.37·15-s + 35/4·16-s − 0.328·17-s + 3.14·18-s + 0.639·19-s − 8.94·20-s + 0.0415·21-s + 3.02·22-s − 0.834·23-s + 5.44·24-s + 2·25-s + 1.20·26-s + 1.58·27-s − 0.269·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(33913.7\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{230} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.06140988351\)
\(L(\frac12)\) \(\approx\) \(0.06140988351\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
23$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 4 T + 46 T^{2} + p^{4} T^{3} + 1190 T^{4} + p^{7} T^{5} + 46 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + T - 135 T^{2} + 521 p T^{3} + 125756 T^{4} + 521 p^{4} T^{5} - 135 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 39 T + 323 p T^{2} + 61307 T^{3} + 4937428 T^{4} + 61307 p^{3} T^{5} + 323 p^{7} T^{6} + 39 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 20 T + 3840 T^{2} + 157301 T^{3} + 10278344 T^{4} + 157301 p^{3} T^{5} + 3840 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 23 T + 5447 T^{2} - 251497 T^{3} + 6295268 T^{4} - 251497 p^{3} T^{5} + 5447 p^{6} T^{6} + 23 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 53 T + 653 p T^{2} - 750501 T^{3} + 76029128 T^{4} - 750501 p^{3} T^{5} + 653 p^{7} T^{6} - 53 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 161 T + 2424 p T^{2} - 9893175 T^{3} + 2240848710 T^{4} - 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} - 161 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 388 T + 132240 T^{2} - 29130015 T^{3} + 5707089166 T^{4} - 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} - 388 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 466 T + 240532 T^{2} - 71494750 T^{3} + 19227960950 T^{4} - 71494750 p^{3} T^{5} + 240532 p^{6} T^{6} - 466 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 484 T + 239452 T^{2} - 82501249 T^{3} + 25012433496 T^{4} - 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 894 T + 484780 T^{2} - 190414350 T^{3} + 59052057270 T^{4} - 190414350 p^{3} T^{5} + 484780 p^{6} T^{6} - 894 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 265 T + 184902 T^{2} + 39015297 T^{3} + 18141867922 T^{4} + 39015297 p^{3} T^{5} + 184902 p^{6} T^{6} + 265 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 576 T + 488156 T^{2} - 177483648 T^{3} + 97785187798 T^{4} - 177483648 p^{3} T^{5} + 488156 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 94 T + 578056 T^{2} + 37319902 T^{3} + 164753482782 T^{4} + 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1472 T + 1805084 T^{2} + 1370871744 T^{3} + 894579577654 T^{4} + 1370871744 p^{3} T^{5} + 1805084 p^{6} T^{6} + 1472 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 200 T + 364322 T^{2} - 121468265 T^{3} + 278790401066 T^{4} - 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1147 T + 1912188 T^{2} - 1347657401 T^{3} + 1180324718270 T^{4} - 1347657401 p^{3} T^{5} + 1912188 p^{6} T^{6} - 1147 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 908 T + 1048492 T^{2} + 1070426012 T^{3} + 693505308902 T^{4} + 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} + 908 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1048 T + 1927704 T^{2} + 1122552888 T^{3} + 1371917773086 T^{4} + 1122552888 p^{3} T^{5} + 1927704 p^{6} T^{6} + 1048 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1784 T + 2327792 T^{2} + 1523808488 T^{3} + 1318662054974 T^{4} + 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} + 1784 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2047 T + 2933893 T^{2} + 2857813609 T^{3} + 3031159147844 T^{4} + 2857813609 p^{3} T^{5} + 2933893 p^{6} T^{6} + 2047 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.517125650491064309584299678751, −8.003092962994205764699342495715, −7.987328015995729355076230620203, −7.916229632306941277190238852518, −7.49205151837231804454471851439, −7.43779176673770000713769677336, −6.87539608483631960585577945622, −6.66228505681297785097783241915, −6.64892521124386147676795191987, −5.96938972579766822693923211827, −5.89214661485449211643328953521, −5.52506133678500379331648376214, −5.42243924147952364428299570897, −4.68229888209149143825694265876, −4.45471270430794561664189539621, −3.97657288497232002418100685636, −3.91517964155954548296894292571, −2.85032150788752427345480824765, −2.70426574251739013494871869266, −2.69184090756482823469013641294, −2.51179737146138258283283578057, −1.34557874092148481925551331671, −0.841714331942519847881041423366, −0.76139953237893540068576956796, −0.14443467134602199433264596732, 0.14443467134602199433264596732, 0.76139953237893540068576956796, 0.841714331942519847881041423366, 1.34557874092148481925551331671, 2.51179737146138258283283578057, 2.69184090756482823469013641294, 2.70426574251739013494871869266, 2.85032150788752427345480824765, 3.91517964155954548296894292571, 3.97657288497232002418100685636, 4.45471270430794561664189539621, 4.68229888209149143825694265876, 5.42243924147952364428299570897, 5.52506133678500379331648376214, 5.89214661485449211643328953521, 5.96938972579766822693923211827, 6.64892521124386147676795191987, 6.66228505681297785097783241915, 6.87539608483631960585577945622, 7.43779176673770000713769677336, 7.49205151837231804454471851439, 7.916229632306941277190238852518, 7.987328015995729355076230620203, 8.003092962994205764699342495715, 8.517125650491064309584299678751

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