Properties

Label 4-450468-1.1-c1e2-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 6·7-s + 9-s + 12-s + 4·13-s + 16-s + 4·19-s − 6·21-s − 2·25-s − 27-s − 6·28-s − 4·31-s − 36-s + 2·37-s − 4·39-s + 9·43-s − 48-s + 14·49-s − 4·52-s − 4·57-s + 14·61-s + 6·63-s − 64-s − 8·67-s + 4·73-s + 2·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 2.26·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.917·19-s − 1.30·21-s − 2/5·25-s − 0.192·27-s − 1.13·28-s − 0.718·31-s − 1/6·36-s + 0.328·37-s − 0.640·39-s + 1.37·43-s − 0.144·48-s + 2·49-s − 0.554·52-s − 0.529·57-s + 1.79·61-s + 0.755·63-s − 1/8·64-s − 0.977·67-s + 0.468·73-s + 0.230·75-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: \(0\)
Selberg data: \((4,\ 450468,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068931227\)
\(L(\frac12)\) \(\approx\) \(2.068931227\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 + T \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 8 T + p T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
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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

−8.421614509791189962331797188341, −8.170186998359684926184999684069, −7.78998041890607523313507624663, −7.23765243145299859028804458710, −6.89889712699493438979408788803, −5.97283179263291363856126743387, −5.69701717715504038134108423394, −5.34000748952416938332431301890, −4.71381624069530879678754181621, −4.39385223217486358221500322685, −3.87836613121471117170905635911, −3.18774593412511201265229334705, −2.16731770562671379526911136441, −1.52333670144635464954712421481, −0.917223546182774170774519678844, 0.917223546182774170774519678844, 1.52333670144635464954712421481, 2.16731770562671379526911136441, 3.18774593412511201265229334705, 3.87836613121471117170905635911, 4.39385223217486358221500322685, 4.71381624069530879678754181621, 5.34000748952416938332431301890, 5.69701717715504038134108423394, 5.97283179263291363856126743387, 6.89889712699493438979408788803, 7.23765243145299859028804458710, 7.78998041890607523313507624663, 8.170186998359684926184999684069, 8.421614509791189962331797188341

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