Properties

Label 4-450468-1.1-c1e2-0-1
Degree $4$
Conductor $450468$
Sign $-1$
Analytic cond. $28.7222$
Root an. cond. $2.31501$
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  − 3-s + 4-s + 9-s − 12-s − 6·13-s + 16-s + 4·19-s − 2·25-s − 27-s + 36-s − 4·37-s + 6·39-s + 13·43-s − 48-s − 14·49-s − 6·52-s − 4·57-s − 22·61-s + 64-s + 4·67-s + 8·73-s + 2·75-s + 4·76-s + 81-s − 11·97-s − 2·100-s − 12·103-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s + 0.917·19-s − 2/5·25-s − 0.192·27-s + 1/6·36-s − 0.657·37-s + 0.960·39-s + 1.98·43-s − 0.144·48-s − 2·49-s − 0.832·52-s − 0.529·57-s − 2.81·61-s + 1/8·64-s + 0.488·67-s + 0.936·73-s + 0.230·75-s + 0.458·76-s + 1/9·81-s − 1.11·97-s − 1/5·100-s − 1.18·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(450468\)    =    \(2^{2} \cdot 3^{3} \cdot 43 \cdot 97\)
Sign: $-1$
Analytic conductor: \(28.7222\)
Root analytic conductor: \(2.31501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{450468} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 450468,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 12 T + p T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 10 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
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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

−8.138832278295180988992082641359, −7.79380793231723333815239822508, −7.33232044179123697279254900974, −7.14118420978245559693474438413, −6.39540539772989034460122647747, −6.14504142692034810129969354879, −5.45233820964288072326165090096, −5.09762044448004847739559428121, −4.63290052093302179442069274802, −4.02929678652284225029587421276, −3.27110713302371014006833787942, −2.73680352755498545831319167822, −2.07121897929449582590926347310, −1.26307831324452862795634492913, 0, 1.26307831324452862795634492913, 2.07121897929449582590926347310, 2.73680352755498545831319167822, 3.27110713302371014006833787942, 4.02929678652284225029587421276, 4.63290052093302179442069274802, 5.09762044448004847739559428121, 5.45233820964288072326165090096, 6.14504142692034810129969354879, 6.39540539772989034460122647747, 7.14118420978245559693474438413, 7.33232044179123697279254900974, 7.79380793231723333815239822508, 8.138832278295180988992082641359

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