# Properties

 Label 10-76e5-1.1-c7e5-0-0 Degree $10$ Conductor $2535525376$ Sign $-1$ Analytic cond. $7.54256\times 10^{6}$ Root an. cond. $4.87250$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $5$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 14·3-s − 280·5-s + 414·7-s − 3.48e3·9-s − 2.66e3·11-s − 602·13-s + 3.92e3·15-s − 2.73e4·17-s − 3.42e4·19-s − 5.79e3·21-s − 6.70e4·23-s − 2.10e5·25-s + 248·27-s − 3.72e5·29-s − 2.71e5·31-s + 3.72e4·33-s − 1.15e5·35-s − 5.62e5·37-s + 8.42e3·39-s − 9.56e5·41-s − 8.27e5·43-s + 9.74e5·45-s − 1.81e6·47-s − 2.40e6·49-s + 3.83e5·51-s + 4.86e5·53-s + 7.45e5·55-s + ⋯
 L(s)  = 1 − 0.299·3-s − 1.00·5-s + 0.456·7-s − 1.59·9-s − 0.603·11-s − 0.0759·13-s + 0.299·15-s − 1.35·17-s − 1.14·19-s − 0.136·21-s − 1.14·23-s − 2.69·25-s + 0.00242·27-s − 2.83·29-s − 1.63·31-s + 0.180·33-s − 0.457·35-s − 1.82·37-s + 0.0227·39-s − 2.16·41-s − 1.58·43-s + 1.59·45-s − 2.54·47-s − 2.91·49-s + 0.404·51-s + 0.449·53-s + 0.604·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{10} \cdot 19^{5}$$ Sign: $-1$ Analytic conductor: $$7.54256\times 10^{6}$$ Root analytic conductor: $$4.87250$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{76} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$5$$ Selberg data: $$(10,\ 2^{10} \cdot 19^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ -1 )$$

## 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 $$1$$
19$C_1$ $$( 1 + p^{3} T )^{5}$$
good3$C_2 \wr S_5$ $$1 + 14 T + 3676 T^{2} + 11104 p^{2} T^{3} + 247145 p^{3} T^{4} + 9484444 p^{3} T^{5} + 247145 p^{10} T^{6} + 11104 p^{16} T^{7} + 3676 p^{21} T^{8} + 14 p^{28} T^{9} + p^{35} T^{10}$$
5$C_2 \wr S_5$ $$1 + 56 p T + 57814 p T^{2} + 2652558 p^{2} T^{3} + 1563182069 p^{2} T^{4} + 55600139212 p^{3} T^{5} + 1563182069 p^{9} T^{6} + 2652558 p^{16} T^{7} + 57814 p^{22} T^{8} + 56 p^{29} T^{9} + p^{35} T^{10}$$
7$C_2 \wr S_5$ $$1 - 414 T + 2575571 T^{2} - 2009518362 T^{3} + 2876468268073 T^{4} - 2752968688370496 T^{5} + 2876468268073 p^{7} T^{6} - 2009518362 p^{14} T^{7} + 2575571 p^{21} T^{8} - 414 p^{28} T^{9} + p^{35} T^{10}$$
11$C_2 \wr S_5$ $$1 + 2 p^{3} T + 47532404 T^{2} + 196653658836 T^{3} + 1564587459786083 T^{4} + 4588703502231464828 T^{5} + 1564587459786083 p^{7} T^{6} + 196653658836 p^{14} T^{7} + 47532404 p^{21} T^{8} + 2 p^{31} T^{9} + p^{35} T^{10}$$
13$C_2 \wr S_5$ $$1 + 602 T + 147508536 T^{2} + 786462907278 T^{3} + 9885416966268075 T^{4} + 88572120386284541784 T^{5} + 9885416966268075 p^{7} T^{6} + 786462907278 p^{14} T^{7} + 147508536 p^{21} T^{8} + 602 p^{28} T^{9} + p^{35} T^{10}$$
17$C_2 \wr S_5$ $$1 + 27366 T + 1416771931 T^{2} + 30995036043348 T^{3} + 976315396052141173 T^{4} +$$$$16\!\cdots\!54$$$$T^{5} + 976315396052141173 p^{7} T^{6} + 30995036043348 p^{14} T^{7} + 1416771931 p^{21} T^{8} + 27366 p^{28} T^{9} + p^{35} T^{10}$$
23$C_2 \wr S_5$ $$1 + 67096 T + 672690382 p T^{2} + 742746864613344 T^{3} + 97403622365062360565 T^{4} +$$$$15\!\cdots\!64$$$$p T^{5} + 97403622365062360565 p^{7} T^{6} + 742746864613344 p^{14} T^{7} + 672690382 p^{22} T^{8} + 67096 p^{28} T^{9} + p^{35} T^{10}$$
29$C_2 \wr S_5$ $$1 + 372398 T + 123594633620 T^{2} + 24322905221765418 T^{3} +$$$$45\!\cdots\!35$$$$T^{4} +$$$$60\!\cdots\!48$$$$T^{5} +$$$$45\!\cdots\!35$$$$p^{7} T^{6} + 24322905221765418 p^{14} T^{7} + 123594633620 p^{21} T^{8} + 372398 p^{28} T^{9} + p^{35} T^{10}$$
31$C_2 \wr S_5$ $$1 + 271372 T + 112879086507 T^{2} + 19060065197093520 T^{3} +$$$$49\!\cdots\!98$$$$T^{4} +$$$$64\!\cdots\!92$$$$T^{5} +$$$$49\!\cdots\!98$$$$p^{7} T^{6} + 19060065197093520 p^{14} T^{7} + 112879086507 p^{21} T^{8} + 271372 p^{28} T^{9} + p^{35} T^{10}$$
37$C_2 \wr S_5$ $$1 + 562630 T + 503027979609 T^{2} + 5044147202230440 p T^{3} +$$$$94\!\cdots\!90$$$$T^{4} +$$$$25\!\cdots\!60$$$$T^{5} +$$$$94\!\cdots\!90$$$$p^{7} T^{6} + 5044147202230440 p^{15} T^{7} + 503027979609 p^{21} T^{8} + 562630 p^{28} T^{9} + p^{35} T^{10}$$
41$C_2 \wr S_5$ $$1 + 956714 T + 858829677401 T^{2} + 558274454732676000 T^{3} +$$$$30\!\cdots\!54$$$$T^{4} +$$$$14\!\cdots\!52$$$$T^{5} +$$$$30\!\cdots\!54$$$$p^{7} T^{6} + 558274454732676000 p^{14} T^{7} + 858829677401 p^{21} T^{8} + 956714 p^{28} T^{9} + p^{35} T^{10}$$
43$C_2 \wr S_5$ $$1 + 827362 T + 1446200884344 T^{2} + 842028213283761024 T^{3} +$$$$80\!\cdots\!43$$$$T^{4} +$$$$33\!\cdots\!88$$$$T^{5} +$$$$80\!\cdots\!43$$$$p^{7} T^{6} + 842028213283761024 p^{14} T^{7} + 1446200884344 p^{21} T^{8} + 827362 p^{28} T^{9} + p^{35} T^{10}$$
47$C_2 \wr S_5$ $$1 + 1812982 T + 2671332480764 T^{2} + 2520094723128764352 T^{3} +$$$$23\!\cdots\!95$$$$T^{4} +$$$$16\!\cdots\!84$$$$T^{5} +$$$$23\!\cdots\!95$$$$p^{7} T^{6} + 2520094723128764352 p^{14} T^{7} + 2671332480764 p^{21} T^{8} + 1812982 p^{28} T^{9} + p^{35} T^{10}$$
53$C_2 \wr S_5$ $$1 - 486998 T + 4629026089784 T^{2} - 1938463029671914986 T^{3} +$$$$98\!\cdots\!83$$$$T^{4} -$$$$32\!\cdots\!92$$$$T^{5} +$$$$98\!\cdots\!83$$$$p^{7} T^{6} - 1938463029671914986 p^{14} T^{7} + 4629026089784 p^{21} T^{8} - 486998 p^{28} T^{9} + p^{35} T^{10}$$
59$C_2 \wr S_5$ $$1 + 367182 T - 1643557954748 T^{2} - 6627210193123102656 T^{3} +$$$$41\!\cdots\!31$$$$T^{4} +$$$$14\!\cdots\!44$$$$T^{5} +$$$$41\!\cdots\!31$$$$p^{7} T^{6} - 6627210193123102656 p^{14} T^{7} - 1643557954748 p^{21} T^{8} + 367182 p^{28} T^{9} + p^{35} T^{10}$$
61$C_2 \wr S_5$ $$1 - 1879732 T + 7749404395290 T^{2} - 9963704865691638198 T^{3} +$$$$38\!\cdots\!65$$$$T^{4} -$$$$47\!\cdots\!92$$$$T^{5} +$$$$38\!\cdots\!65$$$$p^{7} T^{6} - 9963704865691638198 p^{14} T^{7} + 7749404395290 p^{21} T^{8} - 1879732 p^{28} T^{9} + p^{35} T^{10}$$
67$C_2 \wr S_5$ $$1 + 1046394 T + 10097473571576 T^{2} + 25105239457925557932 T^{3} +$$$$64\!\cdots\!63$$$$T^{4} +$$$$25\!\cdots\!36$$$$T^{5} +$$$$64\!\cdots\!63$$$$p^{7} T^{6} + 25105239457925557932 p^{14} T^{7} + 10097473571576 p^{21} T^{8} + 1046394 p^{28} T^{9} + p^{35} T^{10}$$
71$C_2 \wr S_5$ $$1 + 4664572 T + 12023820502163 T^{2} + 22827464116687888320 T^{3} +$$$$99\!\cdots\!02$$$$T^{4} +$$$$47\!\cdots\!84$$$$T^{5} +$$$$99\!\cdots\!02$$$$p^{7} T^{6} + 22827464116687888320 p^{14} T^{7} + 12023820502163 p^{21} T^{8} + 4664572 p^{28} T^{9} + p^{35} T^{10}$$
73$C_2 \wr S_5$ $$1 - 4224942 T + 41896684548731 T^{2} -$$$$13\!\cdots\!28$$$$T^{3} +$$$$81\!\cdots\!65$$$$T^{4} -$$$$20\!\cdots\!14$$$$T^{5} +$$$$81\!\cdots\!65$$$$p^{7} T^{6} -$$$$13\!\cdots\!28$$$$p^{14} T^{7} + 41896684548731 p^{21} T^{8} - 4224942 p^{28} T^{9} + p^{35} T^{10}$$
79$C_2 \wr S_5$ $$1 - 9574024 T + 1156078839417 p T^{2} -$$$$53\!\cdots\!00$$$$T^{3} +$$$$29\!\cdots\!38$$$$T^{4} -$$$$13\!\cdots\!04$$$$T^{5} +$$$$29\!\cdots\!38$$$$p^{7} T^{6} -$$$$53\!\cdots\!00$$$$p^{14} T^{7} + 1156078839417 p^{22} T^{8} - 9574024 p^{28} T^{9} + p^{35} T^{10}$$
83$C_2 \wr S_5$ $$1 - 11754804 T + 110268615554371 T^{2} -$$$$80\!\cdots\!76$$$$T^{3} +$$$$51\!\cdots\!90$$$$T^{4} -$$$$27\!\cdots\!08$$$$T^{5} +$$$$51\!\cdots\!90$$$$p^{7} T^{6} -$$$$80\!\cdots\!76$$$$p^{14} T^{7} + 110268615554371 p^{21} T^{8} - 11754804 p^{28} T^{9} + p^{35} T^{10}$$
89$C_2 \wr S_5$ $$1 - 2782542 T + 110940525514693 T^{2} -$$$$23\!\cdots\!64$$$$T^{3} +$$$$78\!\cdots\!14$$$$T^{4} -$$$$16\!\cdots\!36$$$$T^{5} +$$$$78\!\cdots\!14$$$$p^{7} T^{6} -$$$$23\!\cdots\!64$$$$p^{14} T^{7} + 110940525514693 p^{21} T^{8} - 2782542 p^{28} T^{9} + p^{35} T^{10}$$
97$C_2 \wr S_5$ $$1 - 1291574 T + 106423285884609 T^{2} +$$$$22\!\cdots\!16$$$$T^{3} +$$$$91\!\cdots\!50$$$$T^{4} +$$$$61\!\cdots\!12$$$$T^{5} +$$$$91\!\cdots\!50$$$$p^{7} T^{6} +$$$$22\!\cdots\!16$$$$p^{14} T^{7} + 106423285884609 p^{21} T^{8} - 1291574 p^{28} T^{9} + p^{35} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$