Properties

Label 4-900909-1.1-c1e2-0-1
Degree $4$
Conductor $900909$
Sign $1$
Analytic cond. $57.4427$
Root an. cond. $2.75301$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s − 5·7-s + 9-s + 12-s + 2·13-s − 3·16-s − 19-s + 5·21-s − 6·25-s − 27-s + 5·28-s − 7·31-s − 36-s − 18·37-s − 2·39-s − 11·43-s + 3·48-s + 7·49-s − 2·52-s + 57-s − 2·61-s − 5·63-s + 7·64-s − 18·73-s + 6·75-s + 76-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 1.88·7-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 3/4·16-s − 0.229·19-s + 1.09·21-s − 6/5·25-s − 0.192·27-s + 0.944·28-s − 1.25·31-s − 1/6·36-s − 2.95·37-s − 0.320·39-s − 1.67·43-s + 0.433·48-s + 49-s − 0.277·52-s + 0.132·57-s − 0.256·61-s − 0.629·63-s + 7/8·64-s − 2.10·73-s + 0.692·75-s + 0.114·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900909 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900909 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(900909\)    =    \(3^{3} \cdot 61 \cdot 547\)
Sign: $1$
Analytic conductor: \(57.4427\)
Root analytic conductor: \(2.75301\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{900909} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 900909,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( 1 + T \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
547$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 18 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
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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

−7.49237073350230581233937379789, −7.20978513944233741647612295639, −6.83542518317536574034455166237, −6.37251553769968110056532880459, −5.94312862112266378484655543827, −5.65239344467463663512829107980, −4.95384083987042146393105149573, −4.59878848119341451497687708150, −3.78815627119911322435819474108, −3.53843847648483578811915208480, −3.14325815350386291216495559957, −2.15537093406249369649011088106, −1.53968071585006930556238153242, 0, 0, 1.53968071585006930556238153242, 2.15537093406249369649011088106, 3.14325815350386291216495559957, 3.53843847648483578811915208480, 3.78815627119911322435819474108, 4.59878848119341451497687708150, 4.95384083987042146393105149573, 5.65239344467463663512829107980, 5.94312862112266378484655543827, 6.37251553769968110056532880459, 6.83542518317536574034455166237, 7.20978513944233741647612295639, 7.49237073350230581233937379789

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