Properties

Label 4-155952-1.1-c1e2-0-1
Degree $4$
Conductor $155952$
Sign $1$
Analytic cond. $9.94363$
Root an. cond. $1.77576$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 6·11-s + 2·12-s − 12·13-s − 4·16-s − 2·18-s + 12·22-s + 8·23-s − 9·25-s + 24·26-s + 27-s + 8·32-s − 6·33-s + 2·36-s + 16·37-s − 12·39-s − 12·44-s − 16·46-s + 6·47-s − 4·48-s − 5·49-s + 18·50-s − 24·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 3.32·13-s − 16-s − 0.471·18-s + 2.55·22-s + 1.66·23-s − 9/5·25-s + 4.70·26-s + 0.192·27-s + 1.41·32-s − 1.04·33-s + 1/3·36-s + 2.63·37-s − 1.92·39-s − 1.80·44-s − 2.35·46-s + 0.875·47-s − 0.577·48-s − 5/7·49-s + 2.54·50-s − 3.32·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(155952\)    =    \(2^{4} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(9.94363\)
Root analytic conductor: \(1.77576\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 155952,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4965118511\)
\(L(\frac12)\) \(\approx\) \(0.4965118511\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.400588261603299107064903686230, −8.892718237653967146997373477022, −8.062798749262723709584757589645, −7.935555959343104225865564799258, −7.42319981131404835915630105083, −7.32016227740693371415400237232, −6.66260307528373962910372616260, −5.74803450606645387288578022188, −5.02993077790447449830764321556, −4.85468716698404797055779902636, −4.12511627512637579479297949689, −2.90182171573806321583393234308, −2.47342224903748840497823197638, −2.13307257724406321246824925475, −0.53708427580882030700657091576, 0.53708427580882030700657091576, 2.13307257724406321246824925475, 2.47342224903748840497823197638, 2.90182171573806321583393234308, 4.12511627512637579479297949689, 4.85468716698404797055779902636, 5.02993077790447449830764321556, 5.74803450606645387288578022188, 6.66260307528373962910372616260, 7.32016227740693371415400237232, 7.42319981131404835915630105083, 7.935555959343104225865564799258, 8.062798749262723709584757589645, 8.892718237653967146997373477022, 9.400588261603299107064903686230

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