Properties

Label 4-294e2-1.1-c3e2-0-13
Degree $4$
Conductor $86436$
Sign $1$
Analytic cond. $300.903$
Root an. cond. $4.16492$
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  + 4·2-s + 6·3-s + 12·4-s + 12·5-s + 24·6-s + 32·8-s + 27·9-s + 48·10-s + 4·11-s + 72·12-s + 48·13-s + 72·15-s + 80·16-s + 132·17-s + 108·18-s + 120·19-s + 144·20-s + 16·22-s − 76·23-s + 192·24-s − 140·25-s + 192·26-s + 108·27-s − 112·29-s + 288·30-s + 432·31-s + 192·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.07·5-s + 1.63·6-s + 1.41·8-s + 9-s + 1.51·10-s + 0.109·11-s + 1.73·12-s + 1.02·13-s + 1.23·15-s + 5/4·16-s + 1.88·17-s + 1.41·18-s + 1.44·19-s + 1.60·20-s + 0.155·22-s − 0.689·23-s + 1.63·24-s − 1.11·25-s + 1.44·26-s + 0.769·27-s − 0.717·29-s + 1.75·30-s + 2.50·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86436\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(300.903\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86436,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(15.80143921\)
\(L(\frac12)\) \(\approx\) \(15.80143921\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
3$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 12 T + 284 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 2594 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 48 T + 4520 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 132 T + 820 p T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 120 T + 13086 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 76 T + 4970 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 112 T + 6914 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 432 T + 100406 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 280 T + 56106 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 36 T + 105908 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 128 T - 2778 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 264 T - 3418 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 268 T + 241982 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 336 T + 261374 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 504 T + 268248 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 384 T + 596918 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 396 T + 521098 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 312 T + 94320 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 848 T + 985854 T^{2} + 848 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 648 T + 1235750 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 612 T + 1442324 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 2184 T + 2982432 T^{2} + 2184 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.63431445492111782947110748189, −11.45488916804801928308413297573, −10.41994560012669935765581322805, −10.18696159320214253366392511234, −9.693364519733612874073903625990, −9.521968653605101029101622150662, −8.388565409532993945573580177342, −8.378867158923807661111237419131, −7.43259099217586844543019096058, −7.40837382953303083596435349258, −6.24087391930242974285069401901, −6.23609031078095602791884661059, −5.30603657096106093053196688630, −5.29737636886366103661659571085, −3.97743987756946897094859522790, −3.94023225686257041869166495440, −2.91113894298574609775248979637, −2.84659362013044219475649811598, −1.50585793507008848657758447126, −1.46372001555831665839271572771, 1.46372001555831665839271572771, 1.50585793507008848657758447126, 2.84659362013044219475649811598, 2.91113894298574609775248979637, 3.94023225686257041869166495440, 3.97743987756946897094859522790, 5.29737636886366103661659571085, 5.30603657096106093053196688630, 6.23609031078095602791884661059, 6.24087391930242974285069401901, 7.40837382953303083596435349258, 7.43259099217586844543019096058, 8.378867158923807661111237419131, 8.388565409532993945573580177342, 9.521968653605101029101622150662, 9.693364519733612874073903625990, 10.18696159320214253366392511234, 10.41994560012669935765581322805, 11.45488916804801928308413297573, 11.63431445492111782947110748189

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