Properties

Label 4-294e2-1.1-c3e2-0-15
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 $2$

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: induced by $\chi_{294} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 86436,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−11.27890078967850295722266926844, −10.98490610179967236569511807936, −10.33547488382100185905741210456, −10.14930693512627050254379731050, −9.036961367638614178984677172686, −8.881482788131080571472108435132, −7.77898897740144004195363965895, −7.59734249548219087572375146894, −6.90249757990508154268726875831, −6.70277185182823508289829043185, −5.86472909891021527139150189748, −5.65628817233425873590485737802, −4.77098808881376460145599275815, −4.54233451854330371133894481835, −3.80359878757720019177750551303, −3.65716026932862719939478241863, −2.08297972204198610759983918240, −2.04672904305431429835557206459, 0, 0, 2.04672904305431429835557206459, 2.08297972204198610759983918240, 3.65716026932862719939478241863, 3.80359878757720019177750551303, 4.54233451854330371133894481835, 4.77098808881376460145599275815, 5.65628817233425873590485737802, 5.86472909891021527139150189748, 6.70277185182823508289829043185, 6.90249757990508154268726875831, 7.59734249548219087572375146894, 7.77898897740144004195363965895, 8.881482788131080571472108435132, 9.036961367638614178984677172686, 10.14930693512627050254379731050, 10.33547488382100185905741210456, 10.98490610179967236569511807936, 11.27890078967850295722266926844

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