Properties

Label 4-112e3-1.1-c1e2-0-6
Degree $4$
Conductor $1404928$
Sign $1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
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  + 7-s + 2·9-s + 2·11-s − 10·23-s + 8·25-s + 2·29-s + 14·37-s − 10·43-s + 49-s − 12·53-s + 2·63-s − 8·67-s + 8·71-s + 2·77-s − 4·79-s − 5·81-s + 4·99-s + 8·107-s + 26·109-s + 20·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.377·7-s + 2/3·9-s + 0.603·11-s − 2.08·23-s + 8/5·25-s + 0.371·29-s + 2.30·37-s − 1.52·43-s + 1/7·49-s − 1.64·53-s + 0.251·63-s − 0.977·67-s + 0.949·71-s + 0.227·77-s − 0.450·79-s − 5/9·81-s + 0.402·99-s + 0.773·107-s + 2.49·109-s + 1.88·113-s − 0.909·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1404928\)    =    \(2^{12} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(89.5794\)
Root analytic conductor: \(3.07646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1404928,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.418855533\)
\(L(\frac12)\) \(\approx\) \(2.418855533\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
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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

−7.956664309442800493331047630212, −7.58859647739024456683706443913, −7.05010148801106154077244098451, −6.63816887714249518144690975133, −6.14134670929552779051308300303, −5.98993005102216984812083460162, −5.19533750864123085075422006196, −4.72369091466956213083366103735, −4.36796661297590203351328709610, −3.98942824240376064026147735468, −3.27601579230360099330992906539, −2.81990333719384775133769833012, −1.98868925761822168438950328796, −1.55249081098435501810838380656, −0.70304746349131344247647173726, 0.70304746349131344247647173726, 1.55249081098435501810838380656, 1.98868925761822168438950328796, 2.81990333719384775133769833012, 3.27601579230360099330992906539, 3.98942824240376064026147735468, 4.36796661297590203351328709610, 4.72369091466956213083366103735, 5.19533750864123085075422006196, 5.98993005102216984812083460162, 6.14134670929552779051308300303, 6.63816887714249518144690975133, 7.05010148801106154077244098451, 7.58859647739024456683706443913, 7.956664309442800493331047630212

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