Properties

Label 4-1216e2-1.1-c1e2-0-10
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
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  − 3-s + 5-s + 6·7-s − 9-s + 3·11-s − 13-s − 15-s − 8·17-s − 2·19-s − 6·21-s + 7·23-s − 5·25-s + 3·29-s + 10·31-s − 3·33-s + 6·35-s − 6·37-s + 39-s + 8·41-s − 15·43-s − 45-s + 5·47-s + 13·49-s + 8·51-s + 13·53-s + 3·55-s + 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 2.26·7-s − 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.258·15-s − 1.94·17-s − 0.458·19-s − 1.30·21-s + 1.45·23-s − 25-s + 0.557·29-s + 1.79·31-s − 0.522·33-s + 1.01·35-s − 0.986·37-s + 0.160·39-s + 1.24·41-s − 2.28·43-s − 0.149·45-s + 0.729·47-s + 13/7·49-s + 1.12·51-s + 1.78·53-s + 0.404·55-s + 0.264·57-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506155261\)
\(L(\frac12)\) \(\approx\) \(2.506155261\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 13 T + 156 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 36 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 5 T + 136 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 10 T + 38 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−9.921425562901589940143362404457, −9.575906939164834266073419789204, −8.843029389839722896636014799722, −8.818487832315039984289995285059, −8.290837099466879740148326930035, −8.103142064317069914001051227230, −7.45600856960558964340236169379, −6.92550031419923925262283436369, −6.57252096985603075527736830354, −6.33829063977038368175117962003, −5.47284661848185631628299995762, −5.38001367173071177485282582669, −4.66217418734000829038686331613, −4.60916292509951064396535128823, −4.07718997671769238739851566286, −3.34944380474305733188639208898, −2.35216058809248867116473619110, −2.17863654002928259345487141033, −1.48887825491620288011874390771, −0.73962667314684707641375373116, 0.73962667314684707641375373116, 1.48887825491620288011874390771, 2.17863654002928259345487141033, 2.35216058809248867116473619110, 3.34944380474305733188639208898, 4.07718997671769238739851566286, 4.60916292509951064396535128823, 4.66217418734000829038686331613, 5.38001367173071177485282582669, 5.47284661848185631628299995762, 6.33829063977038368175117962003, 6.57252096985603075527736830354, 6.92550031419923925262283436369, 7.45600856960558964340236169379, 8.103142064317069914001051227230, 8.290837099466879740148326930035, 8.818487832315039984289995285059, 8.843029389839722896636014799722, 9.575906939164834266073419789204, 9.921425562901589940143362404457

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