Properties

Label 8-570e4-1.1-c5e4-0-3
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $6.98460\times 10^{7}$
Root an. cond. $9.56131$
Motivic weight $5$
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  − 16·2-s + 36·3-s + 160·4-s + 100·5-s − 576·6-s + 268·7-s − 1.28e3·8-s + 810·9-s − 1.60e3·10-s + 338·11-s + 5.76e3·12-s + 1.30e3·13-s − 4.28e3·14-s + 3.60e3·15-s + 8.96e3·16-s + 542·17-s − 1.29e4·18-s + 1.44e3·19-s + 1.60e4·20-s + 9.64e3·21-s − 5.40e3·22-s − 88·23-s − 4.60e4·24-s + 6.25e3·25-s − 2.08e4·26-s + 1.45e4·27-s + 4.28e4·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s + 2.06·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s + 0.842·11-s + 11.5·12-s + 2.13·13-s − 5.84·14-s + 4.13·15-s + 35/4·16-s + 0.454·17-s − 9.42·18-s + 0.917·19-s + 8.94·20-s + 4.77·21-s − 2.38·22-s − 0.0346·23-s − 16.3·24-s + 2·25-s − 6.04·26-s + 3.84·27-s + 10.3·28-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(6.98460\times 10^{7}\)
Root analytic conductor: \(9.56131\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(39.06459287\)
\(L(\frac12)\) \(\approx\) \(39.06459287\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{4} \)
3$C_1$ \( ( 1 - p^{2} T )^{4} \)
5$C_1$ \( ( 1 - p^{2} T )^{4} \)
19$C_1$ \( ( 1 - p^{2} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - 268 T + 50902 T^{2} - 8977064 T^{3} + 1428967202 T^{4} - 8977064 p^{5} T^{5} + 50902 p^{10} T^{6} - 268 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 338 T + 269922 T^{2} - 110307258 T^{3} + 66688722682 T^{4} - 110307258 p^{5} T^{5} + 269922 p^{10} T^{6} - 338 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 1302 T + 1269118 T^{2} - 895067978 T^{3} + 537434904594 T^{4} - 895067978 p^{5} T^{5} + 1269118 p^{10} T^{6} - 1302 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 542 T + 2520668 T^{2} - 53548602 p T^{3} + 281978762262 p T^{4} - 53548602 p^{6} T^{5} + 2520668 p^{10} T^{6} - 542 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 88 T + 7186400 T^{2} - 15291204024 T^{3} + 8716186394142 T^{4} - 15291204024 p^{5} T^{5} + 7186400 p^{10} T^{6} + 88 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 5014 T + 2124574 p T^{2} - 287580286770 T^{3} + 1700046374570514 T^{4} - 287580286770 p^{5} T^{5} + 2124574 p^{11} T^{6} - 5014 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 4962 T + 104254328 T^{2} - 407853713778 T^{3} + 4329315214967502 T^{4} - 407853713778 p^{5} T^{5} + 104254328 p^{10} T^{6} - 4962 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 8022 T + 6688862 p T^{2} - 1475777137002 T^{3} + 25079100324614082 T^{4} - 1475777137002 p^{5} T^{5} + 6688862 p^{11} T^{6} - 8022 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 2764 T + 249322694 T^{2} - 1632515334792 T^{3} + 31106725812996186 T^{4} - 1632515334792 p^{5} T^{5} + 249322694 p^{10} T^{6} - 2764 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 25346 T + 732962106 T^{2} - 11160511913746 T^{3} + 171470229081955978 T^{4} - 11160511913746 p^{5} T^{5} + 732962106 p^{10} T^{6} - 25346 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 10008 T + 648822656 T^{2} - 5238435651912 T^{3} + 210610538916792126 T^{4} - 5238435651912 p^{5} T^{5} + 648822656 p^{10} T^{6} - 10008 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 224 T + 665996288 T^{2} + 16576289045760 T^{3} + 137568305857212366 T^{4} + 16576289045760 p^{5} T^{5} + 665996288 p^{10} T^{6} + 224 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 29654 T + 2451079172 T^{2} - 53584799928294 T^{3} + 2525442592409067942 T^{4} - 53584799928294 p^{5} T^{5} + 2451079172 p^{10} T^{6} - 29654 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 27276 T + 3013858744 T^{2} - 69385767593540 T^{3} + 3668289795219064062 T^{4} - 69385767593540 p^{5} T^{5} + 3013858744 p^{10} T^{6} - 27276 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 26024 T + 4899369516 T^{2} - 88385416322536 T^{3} + 9477670181241697846 T^{4} - 88385416322536 p^{5} T^{5} + 4899369516 p^{10} T^{6} - 26024 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 26940 T + 4614821492 T^{2} + 115688848010220 T^{3} + 10068504844200664182 T^{4} + 115688848010220 p^{5} T^{5} + 4614821492 p^{10} T^{6} + 26940 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 60916 T + 6047709796 T^{2} + 339141266895724 T^{3} + 17251637000618592982 T^{4} + 339141266895724 p^{5} T^{5} + 6047709796 p^{10} T^{6} + 60916 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 34902 T + 11783798540 T^{2} - 303252461190558 T^{3} + 53617569850443048870 T^{4} - 303252461190558 p^{5} T^{5} + 11783798540 p^{10} T^{6} - 34902 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 48430 T + 8088541440 T^{2} + 430673209456374 T^{3} + 45339147155957777902 T^{4} + 430673209456374 p^{5} T^{5} + 8088541440 p^{10} T^{6} + 48430 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 38348 T + 17641780214 T^{2} + 533323445064888 T^{3} + \)\(14\!\cdots\!90\)\( T^{4} + 533323445064888 p^{5} T^{5} + 17641780214 p^{10} T^{6} + 38348 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 45942 T + 25565440046 T^{2} - 846184574898498 T^{3} + \)\(29\!\cdots\!66\)\( T^{4} - 846184574898498 p^{5} T^{5} + 25565440046 p^{10} T^{6} - 45942 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.31144338047617505955717758667, −6.67179033628088889541865553261, −6.50453214492779801457285343888, −6.49469174509440078610707868254, −6.39117350858204798412437045619, −5.65554553337824510103351427267, −5.55851813855243457780253703233, −5.43867897393753459748578897190, −5.24603368369475360784769810786, −4.39649707150869806329010654328, −4.27711143770156740478370330081, −4.13896902160653659490452108211, −3.93840482561963732131821276935, −3.16693787605141757273641782859, −2.93006488273541922985060124379, −2.86305624085896817916105608342, −2.84020804765044859236916856727, −2.08186559535185078822348838916, −1.85016339222958260576121931579, −1.78310336414862252472290422783, −1.72165015650152635579501601491, −1.07597534151201279847204857993, −0.975798987156110298495077573061, −0.891955783496050536116547079658, −0.66643135873693098697815610865, 0.66643135873693098697815610865, 0.891955783496050536116547079658, 0.975798987156110298495077573061, 1.07597534151201279847204857993, 1.72165015650152635579501601491, 1.78310336414862252472290422783, 1.85016339222958260576121931579, 2.08186559535185078822348838916, 2.84020804765044859236916856727, 2.86305624085896817916105608342, 2.93006488273541922985060124379, 3.16693787605141757273641782859, 3.93840482561963732131821276935, 4.13896902160653659490452108211, 4.27711143770156740478370330081, 4.39649707150869806329010654328, 5.24603368369475360784769810786, 5.43867897393753459748578897190, 5.55851813855243457780253703233, 5.65554553337824510103351427267, 6.39117350858204798412437045619, 6.49469174509440078610707868254, 6.50453214492779801457285343888, 6.67179033628088889541865553261, 7.31144338047617505955717758667

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