Properties

Label 4-935e2-1.1-c1e2-0-1
Degree $4$
Conductor $874225$
Sign $-1$
Analytic cond. $55.7413$
Root an. cond. $2.73240$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s − 4·7-s − 3·9-s − 2·15-s − 4·16-s − 2·17-s + 8·21-s − 2·23-s − 4·25-s + 14·27-s − 4·35-s + 6·37-s − 3·45-s + 8·48-s − 2·49-s + 4·51-s + 10·59-s + 12·63-s + 4·69-s + 8·73-s + 8·75-s − 4·80-s − 4·81-s − 2·85-s + 30·89-s − 14·97-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 1.51·7-s − 9-s − 0.516·15-s − 16-s − 0.485·17-s + 1.74·21-s − 0.417·23-s − 4/5·25-s + 2.69·27-s − 0.676·35-s + 0.986·37-s − 0.447·45-s + 1.15·48-s − 2/7·49-s + 0.560·51-s + 1.30·59-s + 1.51·63-s + 0.481·69-s + 0.936·73-s + 0.923·75-s − 0.447·80-s − 4/9·81-s − 0.216·85-s + 3.17·89-s − 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 874225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 874225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(874225\)    =    \(5^{2} \cdot 11^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(55.7413\)
Root analytic conductor: \(2.73240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 874225,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−7.944737559627013813143299909880, −7.45089828870553781258206165047, −6.82932770423171918792687431096, −6.36261389471308870138602900888, −6.25340217240276825026401401708, −5.92494984686512651889649464580, −5.34691773503533195072249387087, −4.90231554728495946340439652364, −4.41251199030068389977307598157, −3.59272694843725883717110488971, −3.24206644014110349381627433113, −2.47213473405432371888881268061, −2.13080330824939579119967005255, −0.70848204864155434867787075714, 0, 0.70848204864155434867787075714, 2.13080330824939579119967005255, 2.47213473405432371888881268061, 3.24206644014110349381627433113, 3.59272694843725883717110488971, 4.41251199030068389977307598157, 4.90231554728495946340439652364, 5.34691773503533195072249387087, 5.92494984686512651889649464580, 6.25340217240276825026401401708, 6.36261389471308870138602900888, 6.82932770423171918792687431096, 7.45089828870553781258206165047, 7.944737559627013813143299909880

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