Properties

Label 4-442368-1.1-c1e2-0-5
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·13-s + 8·23-s − 2·25-s − 27-s + 4·37-s + 4·39-s + 8·47-s + 6·49-s + 8·59-s + 4·61-s − 8·69-s + 8·71-s − 4·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s − 8·107-s − 20·109-s − 4·111-s − 4·117-s − 22·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.66·23-s − 2/5·25-s − 0.192·27-s + 0.657·37-s + 0.640·39-s + 1.16·47-s + 6/7·49-s + 1.04·59-s + 0.512·61-s − 0.963·69-s + 0.949·71-s − 0.468·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s − 0.773·107-s − 1.91·109-s − 0.379·111-s − 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{442368} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300683416\)
\(L(\frac12)\) \(\approx\) \(1.300683416\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.620648964698798042027995861180, −8.069296003024719182286593042392, −7.59973622177298338712485066709, −7.11596934108936843626981527026, −6.85707424804806269299182215242, −6.33553372404088741939001164645, −5.61573405155964112907825761587, −5.32453342739906912265969317730, −4.93163605770324517672365246049, −4.14022431258350108091479787197, −3.94743574841707661451444935200, −2.81721097613211638420411402400, −2.62083869433478434502766859076, −1.59209964826416918235844898746, −0.65618734072754594901705452172, 0.65618734072754594901705452172, 1.59209964826416918235844898746, 2.62083869433478434502766859076, 2.81721097613211638420411402400, 3.94743574841707661451444935200, 4.14022431258350108091479787197, 4.93163605770324517672365246049, 5.32453342739906912265969317730, 5.61573405155964112907825761587, 6.33553372404088741939001164645, 6.85707424804806269299182215242, 7.11596934108936843626981527026, 7.59973622177298338712485066709, 8.069296003024719182286593042392, 8.620648964698798042027995861180

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