Properties

Label 4-214272-1.1-c1e2-0-1
Degree $4$
Conductor $214272$
Sign $1$
Analytic cond. $13.6621$
Root an. cond. $1.92256$
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  − 3-s + 9-s + 12·19-s − 6·25-s − 27-s + 5·31-s + 4·37-s + 12·43-s − 14·49-s − 12·57-s + 4·61-s − 12·67-s − 8·73-s + 6·75-s + 20·79-s + 81-s − 5·93-s − 4·97-s − 20·103-s − 4·109-s − 4·111-s + 10·121-s + 127-s − 12·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 2.75·19-s − 6/5·25-s − 0.192·27-s + 0.898·31-s + 0.657·37-s + 1.82·43-s − 2·49-s − 1.58·57-s + 0.512·61-s − 1.46·67-s − 0.936·73-s + 0.692·75-s + 2.25·79-s + 1/9·81-s − 0.518·93-s − 0.406·97-s − 1.97·103-s − 0.383·109-s − 0.379·111-s + 0.909·121-s + 0.0887·127-s − 1.05·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 214272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(214272\)    =    \(2^{8} \cdot 3^{3} \cdot 31\)
Sign: $1$
Analytic conductor: \(13.6621\)
Root analytic conductor: \(1.92256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{214272} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 214272,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.448544963\)
\(L(\frac12)\) \(\approx\) \(1.448544963\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 + T \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.321039659323861061097159872571, −8.530484615407473322321301773177, −7.899973351433242250186053370125, −7.64708463579025774976884144695, −7.24102921467505772339821362992, −6.56947839218317451637377764576, −6.08128156063897742710474004614, −5.59300252499016180443370297334, −5.19183088818837598274241663429, −4.57024261949286174732621188594, −4.00423209001720156661471362856, −3.26085305245353197687960137611, −2.75436860305933135872352463064, −1.68775166527744861991585850623, −0.829285391609422609333888963662, 0.829285391609422609333888963662, 1.68775166527744861991585850623, 2.75436860305933135872352463064, 3.26085305245353197687960137611, 4.00423209001720156661471362856, 4.57024261949286174732621188594, 5.19183088818837598274241663429, 5.59300252499016180443370297334, 6.08128156063897742710474004614, 6.56947839218317451637377764576, 7.24102921467505772339821362992, 7.64708463579025774976884144695, 7.899973351433242250186053370125, 8.530484615407473322321301773177, 9.321039659323861061097159872571

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