Properties

Label 8-570e4-1.1-c7e4-0-2
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $1.00521\times 10^{9}$
Root an. cond. $13.3438$
Motivic weight $7$
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  + 32·2-s + 108·3-s + 640·4-s + 500·5-s + 3.45e3·6-s − 1.31e3·7-s + 1.02e4·8-s + 7.29e3·9-s + 1.60e4·10-s − 6.57e3·11-s + 6.91e4·12-s − 1.75e4·13-s − 4.21e4·14-s + 5.40e4·15-s + 1.43e5·16-s − 2.84e4·17-s + 2.33e5·18-s − 2.74e4·19-s + 3.20e5·20-s − 1.42e5·21-s − 2.10e5·22-s − 3.21e4·23-s + 1.10e6·24-s + 1.56e5·25-s − 5.62e5·26-s + 3.93e5·27-s − 8.42e5·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s − 1.45·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s − 1.48·11-s + 11.5·12-s − 2.21·13-s − 4.10·14-s + 4.13·15-s + 35/4·16-s − 1.40·17-s + 9.42·18-s − 0.917·19-s + 8.94·20-s − 3.34·21-s − 4.20·22-s − 0.550·23-s + 16.3·24-s + 2·25-s − 6.27·26-s + 3.84·27-s − 7.25·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.00521\times 10^{9}\)
Root analytic conductor: \(13.3438\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{4} \)
3$C_1$ \( ( 1 - p^{3} T )^{4} \)
5$C_1$ \( ( 1 - p^{3} T )^{4} \)
19$C_1$ \( ( 1 + p^{3} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 188 p T + 3092668 T^{2} + 398881876 p T^{3} + 3643191635462 T^{4} + 398881876 p^{8} T^{5} + 3092668 p^{14} T^{6} + 188 p^{22} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 6570 T + 55802072 T^{2} + 148633201962 T^{3} + 992291347332990 T^{4} + 148633201962 p^{7} T^{5} + 55802072 p^{14} T^{6} + 6570 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 17578 T + 221418772 T^{2} + 1876174025262 T^{3} + 15069896273646150 T^{4} + 1876174025262 p^{7} T^{5} + 221418772 p^{14} T^{6} + 17578 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 28466 T + 837051992 T^{2} + 8365939855630 T^{3} + 229175463710842190 T^{4} + 8365939855630 p^{7} T^{5} + 837051992 p^{14} T^{6} + 28466 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 32136 T + 7488127196 T^{2} + 510556935133352 T^{3} + 26186381560297811238 T^{4} + 510556935133352 p^{7} T^{5} + 7488127196 p^{14} T^{6} + 32136 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 159122 T + 64719167228 T^{2} + 6654469087470278 T^{3} + \)\(15\!\cdots\!62\)\( T^{4} + 6654469087470278 p^{7} T^{5} + 64719167228 p^{14} T^{6} + 159122 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 67974 T + 58226470540 T^{2} - 1806216343105522 T^{3} + \)\(14\!\cdots\!74\)\( T^{4} - 1806216343105522 p^{7} T^{5} + 58226470540 p^{14} T^{6} + 67974 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 823702 T + 490118548108 T^{2} + 216128024231822626 T^{3} + \)\(75\!\cdots\!18\)\( T^{4} + 216128024231822626 p^{7} T^{5} + 490118548108 p^{14} T^{6} + 823702 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 781924 T + 767506780388 T^{2} + 382623949748239252 T^{3} + \)\(22\!\cdots\!30\)\( T^{4} + 382623949748239252 p^{7} T^{5} + 767506780388 p^{14} T^{6} + 781924 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 1115638 T + 737266652992 T^{2} + 356238832104359286 T^{3} + \)\(19\!\cdots\!38\)\( T^{4} + 356238832104359286 p^{7} T^{5} + 737266652992 p^{14} T^{6} + 1115638 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 209160 T + 863688190268 T^{2} + 197548391176963560 T^{3} + \)\(56\!\cdots\!98\)\( T^{4} + 197548391176963560 p^{7} T^{5} + 863688190268 p^{14} T^{6} + 209160 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 16008 p T + 4377691198700 T^{2} + 2779662163460513976 T^{3} + \)\(75\!\cdots\!54\)\( T^{4} + 2779662163460513976 p^{7} T^{5} + 4377691198700 p^{14} T^{6} + 16008 p^{22} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 3677830 T + 12144428527772 T^{2} + 27009966703230094102 T^{3} + \)\(48\!\cdots\!34\)\( T^{4} + 27009966703230094102 p^{7} T^{5} + 12144428527772 p^{14} T^{6} + 3677830 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1161072 T + 7069720468828 T^{2} + 313029995829507664 T^{3} + \)\(19\!\cdots\!70\)\( T^{4} + 313029995829507664 p^{7} T^{5} + 7069720468828 p^{14} T^{6} + 1161072 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 6154740 T + 22793858868268 T^{2} + 76558374281921978036 T^{3} + \)\(21\!\cdots\!34\)\( T^{4} + 76558374281921978036 p^{7} T^{5} + 22793858868268 p^{14} T^{6} + 6154740 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 4456224 T + 35224215958364 T^{2} + \)\(11\!\cdots\!28\)\( T^{3} + \)\(47\!\cdots\!26\)\( T^{4} + \)\(11\!\cdots\!28\)\( p^{7} T^{5} + 35224215958364 p^{14} T^{6} + 4456224 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1057792 T + 25051579433164 T^{2} - 23683837938073896192 T^{3} + \)\(38\!\cdots\!02\)\( T^{4} - 23683837938073896192 p^{7} T^{5} + 25051579433164 p^{14} T^{6} - 1057792 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 2910090 T + 76806351838396 T^{2} - \)\(16\!\cdots\!98\)\( T^{3} + \)\(22\!\cdots\!26\)\( T^{4} - \)\(16\!\cdots\!98\)\( p^{7} T^{5} + 76806351838396 p^{14} T^{6} - 2910090 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1767198 T + 77723349910148 T^{2} + \)\(19\!\cdots\!66\)\( T^{3} + \)\(27\!\cdots\!98\)\( T^{4} + \)\(19\!\cdots\!66\)\( p^{7} T^{5} + 77723349910148 p^{14} T^{6} + 1767198 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3677360 T + 91147101194132 T^{2} - \)\(13\!\cdots\!48\)\( T^{3} + \)\(30\!\cdots\!94\)\( T^{4} - \)\(13\!\cdots\!48\)\( p^{7} T^{5} + 91147101194132 p^{14} T^{6} + 3677360 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 10419094 T + 332866439705620 T^{2} + \)\(23\!\cdots\!54\)\( T^{3} + \)\(40\!\cdots\!70\)\( T^{4} + \)\(23\!\cdots\!54\)\( p^{7} T^{5} + 332866439705620 p^{14} T^{6} + 10419094 p^{21} T^{7} + p^{28} T^{8} \)
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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

−7.09179314985610951217750757953, −6.52610430234439246422358763149, −6.46227586801924299683653222031, −6.40493793659401919617213661217, −6.36422169591260170142825422954, −5.59607989672739009182414260566, −5.49028853716415767026723367132, −5.29671369107858346089021260590, −5.20362059789404779297768638778, −4.67560936424988803909336313840, −4.66334601038369614699235873604, −4.39055568691307006003267090815, −4.29185458209951853965887565189, −3.62182470302407366459427218627, −3.43318808689292782663253766660, −3.27730108804324398844902894014, −3.20202485268552025394665118929, −2.74262283894865553712693231409, −2.68626070161235747802445574536, −2.35666244406764329841354161024, −2.34121246871538554542426648285, −1.73513966362820399082891016570, −1.68208977321007778057984002509, −1.55535584712508428876808345699, −1.49482749451068041985197332545, 0, 0, 0, 0, 1.49482749451068041985197332545, 1.55535584712508428876808345699, 1.68208977321007778057984002509, 1.73513966362820399082891016570, 2.34121246871538554542426648285, 2.35666244406764329841354161024, 2.68626070161235747802445574536, 2.74262283894865553712693231409, 3.20202485268552025394665118929, 3.27730108804324398844902894014, 3.43318808689292782663253766660, 3.62182470302407366459427218627, 4.29185458209951853965887565189, 4.39055568691307006003267090815, 4.66334601038369614699235873604, 4.67560936424988803909336313840, 5.20362059789404779297768638778, 5.29671369107858346089021260590, 5.49028853716415767026723367132, 5.59607989672739009182414260566, 6.36422169591260170142825422954, 6.40493793659401919617213661217, 6.46227586801924299683653222031, 6.52610430234439246422358763149, 7.09179314985610951217750757953

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