Properties

Degree $4$
Conductor $63504$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·7-s + 10·13-s + 19-s + 5·25-s − 11·31-s − 11·37-s − 26·43-s + 18·49-s − 14·61-s − 5·67-s − 17·73-s − 17·79-s + 50·91-s + 28·97-s + 13·103-s + 19·109-s + 11·121-s + 127-s + 131-s + 5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.88·7-s + 2.77·13-s + 0.229·19-s + 25-s − 1.97·31-s − 1.80·37-s − 3.96·43-s + 18/7·49-s − 1.79·61-s − 0.610·67-s − 1.98·73-s − 1.91·79-s + 5.24·91-s + 2.84·97-s + 1.28·103-s + 1.81·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.433·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 63504,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94242\)
\(L(\frac12)\) \(\approx\) \(1.94242\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−11.99929450501712945365765238435, −11.76781945541260148277579085538, −11.16649553244544186493239264889, −11.08157249016867279069507421164, −10.37132101945693126350887489407, −10.26770479503104807237374766039, −8.989685743507978045566270894294, −8.891602898263775853725812511253, −8.372007332579292663194765642537, −8.172325327717420668150147378323, −7.13029545854197214861403933751, −7.12169118176407111083321705790, −5.90699610136419423916435487577, −5.89035506414127979385176340765, −4.84025151217890051943393771530, −4.71978140179157600387267003743, −3.46639878577192253208900107072, −3.46622239353313215930627619790, −1.67259348408527152003717601706, −1.54297146000919957214995670622, 1.54297146000919957214995670622, 1.67259348408527152003717601706, 3.46622239353313215930627619790, 3.46639878577192253208900107072, 4.71978140179157600387267003743, 4.84025151217890051943393771530, 5.89035506414127979385176340765, 5.90699610136419423916435487577, 7.12169118176407111083321705790, 7.13029545854197214861403933751, 8.172325327717420668150147378323, 8.372007332579292663194765642537, 8.891602898263775853725812511253, 8.989685743507978045566270894294, 10.26770479503104807237374766039, 10.37132101945693126350887489407, 11.08157249016867279069507421164, 11.16649553244544186493239264889, 11.76781945541260148277579085538, 11.99929450501712945365765238435

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