Properties

Label 4-1216e2-1.1-c3e2-0-3
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 10·5-s − 57·7-s − 9·9-s + 10·11-s − 13·13-s − 10·15-s − 51·17-s − 38·19-s − 57·21-s + 155·23-s + 2·25-s + 8·27-s + 79·29-s + 16·31-s + 10·33-s + 570·35-s − 380·37-s − 13·39-s − 790·41-s + 296·43-s + 90·45-s + 200·47-s + 1.79e3·49-s − 51·51-s − 397·53-s − 100·55-s + ⋯
L(s)  = 1  + 0.192·3-s − 0.894·5-s − 3.07·7-s − 1/3·9-s + 0.274·11-s − 0.277·13-s − 0.172·15-s − 0.727·17-s − 0.458·19-s − 0.592·21-s + 1.40·23-s + 0.0159·25-s + 0.0570·27-s + 0.505·29-s + 0.0926·31-s + 0.0527·33-s + 2.75·35-s − 1.68·37-s − 0.0533·39-s − 3.00·41-s + 1.04·43-s + 0.298·45-s + 0.620·47-s + 5.23·49-s − 0.140·51-s − 1.02·53-s − 0.245·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 - T + 10 T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + 2 p T + 98 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 57 T + 1454 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 10 T + 2510 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + p T + 2268 T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 3 p T + 520 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 155 T + 994 p T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 79 T + 13124 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 16 T + 48318 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 380 T + 126078 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 296 T + 78966 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 200 T + 146846 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 397 T + 333572 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 201 T + 197794 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 680 T + 483894 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 939 T + 740138 T^{2} + 939 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 406 T + 735614 T^{2} + 406 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 123 T + 781772 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 106 T + 840030 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 2226 T + 2380750 T^{2} - 2226 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1864 T + 2438382 T^{2} + 1864 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.115591743301416199302857930347, −8.897181636962605483390146839630, −8.449458603357767068196691145450, −8.027192870226431855214496931993, −7.23296308135166431020012858388, −6.98916443232887175328796850054, −6.74210356246334583246455982095, −6.43675559353954060121423570727, −5.91827891666826328459040700990, −5.37967031715861439540260419790, −4.83885141795867023543641392160, −4.23292679634972068933911494695, −3.63389331023466556412701610115, −3.48102484450201908960288781950, −2.91386327102178614091579413267, −2.67356849369048313064078196400, −1.76574158876416607840065488590, −0.75228481363562075767106516236, 0, 0, 0.75228481363562075767106516236, 1.76574158876416607840065488590, 2.67356849369048313064078196400, 2.91386327102178614091579413267, 3.48102484450201908960288781950, 3.63389331023466556412701610115, 4.23292679634972068933911494695, 4.83885141795867023543641392160, 5.37967031715861439540260419790, 5.91827891666826328459040700990, 6.43675559353954060121423570727, 6.74210356246334583246455982095, 6.98916443232887175328796850054, 7.23296308135166431020012858388, 8.027192870226431855214496931993, 8.449458603357767068196691145450, 8.897181636962605483390146839630, 9.115591743301416199302857930347

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