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Properties

Label 4-14112-1.1-c1e2-0-8
Degree $4$
Conductor $14112$
Sign $-1$
Analytic cond. $0.899793$
Root an. cond. $0.973947$
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  − 2-s + 4-s − 4·7-s − 8-s − 9-s − 8·11-s + 4·14-s + 16-s + 18-s + 8·22-s − 2·25-s − 4·28-s − 4·29-s − 32-s − 36-s − 4·37-s + 8·43-s − 8·44-s + 9·49-s + 2·50-s − 4·53-s + 4·56-s + 4·58-s + 4·63-s + 64-s + 8·67-s + 72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1/3·9-s − 2.41·11-s + 1.06·14-s + 1/4·16-s + 0.235·18-s + 1.70·22-s − 2/5·25-s − 0.755·28-s − 0.742·29-s − 0.176·32-s − 1/6·36-s − 0.657·37-s + 1.21·43-s − 1.20·44-s + 9/7·49-s + 0.282·50-s − 0.549·53-s + 0.534·56-s + 0.525·58-s + 0.503·63-s + 1/8·64-s + 0.977·67-s + 0.117·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14112\)    =    \(2^{5} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(0.899793\)
Root analytic conductor: \(0.973947\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 14112,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
3$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.e_bu
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ai_bm
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.47.a_ade
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.e_dq
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.61.a_abe
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.73.a_as
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ai_eg
83$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.83.a_k
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.97.a_bu
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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

−10.83937199315662484586716469501, −10.31111141594447444617275360518, −9.829304100911114339743068912173, −9.369742229008080646577255900558, −8.764378636510801461127509140453, −7.973750574527371418162507203307, −7.70876721394217607263243069029, −6.96936646197174054580566893474, −6.32252985921631827171222078311, −5.60436688805546147397987493556, −5.16172953442218110258125969526, −3.85972029850219265617674503233, −2.97335643137218365686048893684, −2.38989316675495611504703574176, 0, 2.38989316675495611504703574176, 2.97335643137218365686048893684, 3.85972029850219265617674503233, 5.16172953442218110258125969526, 5.60436688805546147397987493556, 6.32252985921631827171222078311, 6.96936646197174054580566893474, 7.70876721394217607263243069029, 7.973750574527371418162507203307, 8.764378636510801461127509140453, 9.369742229008080646577255900558, 9.829304100911114339743068912173, 10.31111141594447444617275360518, 10.83937199315662484586716469501

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