Properties

Label 4-112e3-1.1-c1e2-0-1
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 − 8·23-s − 6·25-s − 4·29-s − 4·37-s + 49-s + 4·53-s − 2·63-s + 24·67-s + 16·71-s − 5·81-s + 8·107-s − 4·109-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s − 6·169-s + ⋯
L(s)  = 1  − 0.377·7-s + 2/3·9-s − 1.66·23-s − 6/5·25-s − 0.742·29-s − 0.657·37-s + 1/7·49-s + 0.549·53-s − 0.251·63-s + 2.93·67-s + 1.89·71-s − 5/9·81-s + 0.773·107-s − 0.383·109-s − 1.12·113-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 + 0.630·161-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-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

\(L(1)\) \(\approx\) \(1.479234028\)
\(L(\frac12)\) \(\approx\) \(1.479234028\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
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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

−7.891324420127686085550129195909, −7.59323168715670500590973969136, −6.95540896797002597284237909041, −6.78523961215961321298954807630, −6.14275733321769910969643004422, −5.83873273835049553454047245541, −5.30885500505738297133867999554, −4.89821029891924306193690564046, −4.15658020906930397374781788636, −3.77775849182394915622704046945, −3.61175651732633785166651233840, −2.63188894292257056466925258503, −2.10686711822222498297603419961, −1.60826206523482221383180587998, −0.51896389410269347045509565112, 0.51896389410269347045509565112, 1.60826206523482221383180587998, 2.10686711822222498297603419961, 2.63188894292257056466925258503, 3.61175651732633785166651233840, 3.77775849182394915622704046945, 4.15658020906930397374781788636, 4.89821029891924306193690564046, 5.30885500505738297133867999554, 5.83873273835049553454047245541, 6.14275733321769910969643004422, 6.78523961215961321298954807630, 6.95540896797002597284237909041, 7.59323168715670500590973969136, 7.891324420127686085550129195909

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