Properties

Label 4-825e2-1.1-c1e2-0-20
Degree $4$
Conductor $680625$
Sign $-1$
Analytic cond. $43.3972$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s + 2·11-s − 4·16-s − 12·23-s + 4·27-s − 6·31-s + 4·33-s − 4·37-s − 4·47-s − 8·48-s − 5·49-s + 8·53-s − 20·59-s + 6·67-s − 24·69-s − 16·71-s + 5·81-s − 12·93-s − 34·97-s + 6·99-s + 8·103-s − 8·111-s + 8·113-s − 7·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 0.603·11-s − 16-s − 2.50·23-s + 0.769·27-s − 1.07·31-s + 0.696·33-s − 0.657·37-s − 0.583·47-s − 1.15·48-s − 5/7·49-s + 1.09·53-s − 2.60·59-s + 0.733·67-s − 2.88·69-s − 1.89·71-s + 5/9·81-s − 1.24·93-s − 3.45·97-s + 0.603·99-s + 0.788·103-s − 0.759·111-s + 0.752·113-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(43.3972\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 680625,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
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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

−8.114757177440921419069207299350, −7.74300600574245161145141332474, −7.34923234321660315078497154764, −6.77287251257420576381216593643, −6.47550891297445546652062110512, −5.83602188561825449212965209319, −5.44264591099408884558544576394, −4.48018036005395779311676580836, −4.38637866918120839883207984982, −3.75734065175266443138175607180, −3.28631221908682607371983175186, −2.62960787106095443058504699500, −1.90319955654708104595670502735, −1.60835528081174442082226795369, 0, 1.60835528081174442082226795369, 1.90319955654708104595670502735, 2.62960787106095443058504699500, 3.28631221908682607371983175186, 3.75734065175266443138175607180, 4.38637866918120839883207984982, 4.48018036005395779311676580836, 5.44264591099408884558544576394, 5.83602188561825449212965209319, 6.47550891297445546652062110512, 6.77287251257420576381216593643, 7.34923234321660315078497154764, 7.74300600574245161145141332474, 8.114757177440921419069207299350

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