Properties

Degree $4$
Conductor $3418801$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 2·11-s − 6·13-s − 14·17-s + 2·23-s − 4·25-s − 6·31-s − 10·41-s − 4·44-s − 20·47-s − 8·49-s − 12·52-s − 2·53-s − 20·59-s − 8·64-s + 18·67-s − 28·68-s − 12·79-s − 9·81-s + 2·83-s + 4·92-s + 22·97-s − 8·100-s − 10·101-s + 34·103-s + 28·107-s + 26·109-s + ⋯
L(s)  = 1  + 4-s − 0.603·11-s − 1.66·13-s − 3.39·17-s + 0.417·23-s − 4/5·25-s − 1.07·31-s − 1.56·41-s − 0.603·44-s − 2.91·47-s − 8/7·49-s − 1.66·52-s − 0.274·53-s − 2.60·59-s − 64-s + 2.19·67-s − 3.39·68-s − 1.35·79-s − 81-s + 0.219·83-s + 0.417·92-s + 2.23·97-s − 4/5·100-s − 0.995·101-s + 3.35·103-s + 2.70·107-s + 2.49·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3418801\)    =    \(43^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1849} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3418801,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad43 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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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

−9.073381058447805601256365309005, −8.649842777033441119852664742335, −8.337089079911450708883195215285, −7.60584991633589573479032289791, −7.53559212230323752887327208127, −7.04407308951021410917296057822, −6.67863814172410917138263516897, −6.25002369939981202080222924059, −6.21377957399014347954774776352, −5.19349748322025490914723346512, −4.91963650750806191459270266519, −4.66217848583934006532747445107, −4.23292310430270266036539560339, −3.21098522528394051622634132865, −3.19392624763796702691461080328, −2.31122467508602671580007165175, −1.92387577974938266647656403220, −1.88726173175353408736274236765, 0, 0, 1.88726173175353408736274236765, 1.92387577974938266647656403220, 2.31122467508602671580007165175, 3.19392624763796702691461080328, 3.21098522528394051622634132865, 4.23292310430270266036539560339, 4.66217848583934006532747445107, 4.91963650750806191459270266519, 5.19349748322025490914723346512, 6.21377957399014347954774776352, 6.25002369939981202080222924059, 6.67863814172410917138263516897, 7.04407308951021410917296057822, 7.53559212230323752887327208127, 7.60584991633589573479032289791, 8.337089079911450708883195215285, 8.649842777033441119852664742335, 9.073381058447805601256365309005

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