Properties

Label 4-1120e2-1.1-c1e2-0-3
Degree $4$
Conductor $1254400$
Sign $1$
Analytic cond. $79.9816$
Root an. cond. $2.99052$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 6·9-s − 8·23-s − 25-s + 9·49-s − 24·63-s + 27·81-s − 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s + 22·169-s + 173-s + 4·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 1.51·7-s + 2·9-s − 1.66·23-s − 1/5·25-s + 9/7·49-s − 3.02·63-s + 3·81-s − 3.38·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.302·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1254400\)    =    \(2^{10} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(79.9816\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1254400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1254400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.601904542\)
\(L(\frac12)\) \(\approx\) \(1.601904542\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.906220376624073770155864429395, −7.58857818303238706159650002230, −6.93689615691223189738215807520, −6.77905504056966748772240958211, −6.42208896630828348032227290824, −5.81087992542183484016566817702, −5.50780588372728798325001666435, −4.69950754327363854254992746528, −4.27260335026771230676567670892, −3.91766415698140698102902070003, −3.46113607685169582028862835498, −2.84017334500500223097151973196, −2.08465081524935937309819714626, −1.56357322386974514950596517545, −0.56574862696975978705379798295, 0.56574862696975978705379798295, 1.56357322386974514950596517545, 2.08465081524935937309819714626, 2.84017334500500223097151973196, 3.46113607685169582028862835498, 3.91766415698140698102902070003, 4.27260335026771230676567670892, 4.69950754327363854254992746528, 5.50780588372728798325001666435, 5.81087992542183484016566817702, 6.42208896630828348032227290824, 6.77905504056966748772240958211, 6.93689615691223189738215807520, 7.58857818303238706159650002230, 7.906220376624073770155864429395

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