Properties

Degree $4$
Conductor $1806336$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s − 5·7-s − 3·11-s + 8·13-s − 3·15-s + 4·19-s + 5·21-s + 5·25-s + 27-s − 18·29-s − 31-s + 3·33-s − 15·35-s + 8·37-s − 8·39-s − 20·43-s − 6·47-s + 18·49-s − 3·53-s − 9·55-s − 4·57-s − 3·59-s − 10·61-s + 24·65-s + 10·67-s + 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 1.88·7-s − 0.904·11-s + 2.21·13-s − 0.774·15-s + 0.917·19-s + 1.09·21-s + 25-s + 0.192·27-s − 3.34·29-s − 0.179·31-s + 0.522·33-s − 2.53·35-s + 1.31·37-s − 1.28·39-s − 3.04·43-s − 0.875·47-s + 18/7·49-s − 0.412·53-s − 1.21·55-s − 0.529·57-s − 0.390·59-s − 1.28·61-s + 2.97·65-s + 1.22·67-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.175915585\)
\(L(\frac12)\) \(\approx\) \(1.175915585\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.847129592906555769621365340945, −9.617811925300647502603449069179, −9.191027185931119777766546501830, −8.566238297858644265930923755764, −8.460211171309675713827260941914, −7.62411236990218427561809685546, −7.37428904978834398234973024076, −6.64494406104385811027769021964, −6.48730825145589396331878131390, −6.00759532064066943862210745351, −5.80939920583384302638292252103, −5.33894426610144068278579162544, −5.10711785909175866657311552068, −4.06400188275481559516034203982, −3.71443981500927125131399719682, −3.08691029457206792552022307801, −2.97864268332861178571832407954, −1.88787079251985058206090425438, −1.55445830857693967280399678653, −0.45760126575825555831842628499, 0.45760126575825555831842628499, 1.55445830857693967280399678653, 1.88787079251985058206090425438, 2.97864268332861178571832407954, 3.08691029457206792552022307801, 3.71443981500927125131399719682, 4.06400188275481559516034203982, 5.10711785909175866657311552068, 5.33894426610144068278579162544, 5.80939920583384302638292252103, 6.00759532064066943862210745351, 6.48730825145589396331878131390, 6.64494406104385811027769021964, 7.37428904978834398234973024076, 7.62411236990218427561809685546, 8.460211171309675713827260941914, 8.566238297858644265930923755764, 9.191027185931119777766546501830, 9.617811925300647502603449069179, 9.847129592906555769621365340945

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