Properties

Label 4-900909-1.1-c1e2-0-0
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 7-s + 9-s − 12-s − 3·16-s − 7·19-s + 21-s + 8·25-s − 27-s − 28-s + 5·31-s + 36-s − 2·37-s + 3·43-s + 3·48-s − 7·49-s + 7·57-s + 2·61-s − 63-s − 7·64-s + 12·67-s − 6·73-s − 8·75-s − 7·76-s − 9·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.377·7-s + 1/3·9-s − 0.288·12-s − 3/4·16-s − 1.60·19-s + 0.218·21-s + 8/5·25-s − 0.192·27-s − 0.188·28-s + 0.898·31-s + 1/6·36-s − 0.328·37-s + 0.457·43-s + 0.433·48-s − 49-s + 0.927·57-s + 0.256·61-s − 0.125·63-s − 7/8·64-s + 1.46·67-s − 0.702·73-s − 0.923·75-s − 0.802·76-s − 1.01·79-s + 1/9·81-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: \(1\)
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$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
547$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 14 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−8.045888184951816908350796596267, −7.24107758644621914311812206180, −7.00858074215859027940483052403, −6.67274054544685041607888895754, −6.15654972047410318769002504698, −5.96576091623653713998042006209, −5.16838618929762093880028981211, −4.74781859979719160627074585645, −4.36747711423985775090863259107, −3.78151580945457285893508580590, −3.05764083738055862565833930880, −2.54541430326249375963403735429, −1.96666366011643356093659772246, −1.09207602760955137550356277339, 0, 1.09207602760955137550356277339, 1.96666366011643356093659772246, 2.54541430326249375963403735429, 3.05764083738055862565833930880, 3.78151580945457285893508580590, 4.36747711423985775090863259107, 4.74781859979719160627074585645, 5.16838618929762093880028981211, 5.96576091623653713998042006209, 6.15654972047410318769002504698, 6.67274054544685041607888895754, 7.00858074215859027940483052403, 7.24107758644621914311812206180, 8.045888184951816908350796596267

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