Properties

Label 4-945e2-1.1-c1e2-0-62
Degree $4$
Conductor $893025$
Sign $1$
Analytic cond. $56.9400$
Root an. cond. $2.74697$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·5-s + 2·7-s − 8-s + 2·10-s − 6·11-s − 13-s − 2·14-s − 16-s + 17-s − 3·19-s + 6·22-s − 5·23-s + 3·25-s + 26-s − 7·29-s − 12·31-s + 6·32-s − 34-s − 4·35-s + 3·38-s + 2·40-s + 3·41-s + 6·43-s + 5·46-s − 8·47-s + 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.894·5-s + 0.755·7-s − 0.353·8-s + 0.632·10-s − 1.80·11-s − 0.277·13-s − 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.688·19-s + 1.27·22-s − 1.04·23-s + 3/5·25-s + 0.196·26-s − 1.29·29-s − 2.15·31-s + 1.06·32-s − 0.171·34-s − 0.676·35-s + 0.486·38-s + 0.316·40-s + 0.468·41-s + 0.914·43-s + 0.737·46-s − 1.16·47-s + 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(893025\)    =    \(3^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(56.9400\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 893025,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 49 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T - 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 15 T + 169 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 23 T + 261 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 139 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 113 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−9.619636564905851806587219059312, −9.555203656778963903074831960031, −8.753108903507971250433795228050, −8.751746540966947878905354743513, −7.969054625215624680121296890813, −7.88649282747432496796490652381, −7.58391826096975298495845965602, −7.07399584348716094016968446731, −6.54542232984902087688770967563, −5.83083040081622648767598499449, −5.42865668409504728035112464771, −5.14878958425915842536333551003, −4.32000914349525075392100292482, −4.16183705672524791240038023341, −3.41570660879847331405735691002, −2.73692037708999870764562246318, −2.23933210001705691252504392200, −1.51779104103791489479495855848, 0, 0, 1.51779104103791489479495855848, 2.23933210001705691252504392200, 2.73692037708999870764562246318, 3.41570660879847331405735691002, 4.16183705672524791240038023341, 4.32000914349525075392100292482, 5.14878958425915842536333551003, 5.42865668409504728035112464771, 5.83083040081622648767598499449, 6.54542232984902087688770967563, 7.07399584348716094016968446731, 7.58391826096975298495845965602, 7.88649282747432496796490652381, 7.969054625215624680121296890813, 8.751746540966947878905354743513, 8.753108903507971250433795228050, 9.555203656778963903074831960031, 9.619636564905851806587219059312

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