Properties

Label 4-36000-1.1-c1e2-0-4
Degree $4$
Conductor $36000$
Sign $1$
Analytic cond. $2.29539$
Root an. cond. $1.23087$
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  − 2-s + 2·3-s + 4-s − 5-s − 2·6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s + 2·12-s − 2·14-s − 2·15-s + 16-s − 18-s − 20-s + 4·21-s − 6·22-s − 2·24-s + 25-s − 4·27-s + 2·28-s + 2·30-s − 32-s + 12·33-s − 2·35-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.577·12-s − 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s + 0.872·21-s − 1.27·22-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.377·28-s + 0.365·30-s − 0.176·32-s + 2.08·33-s − 0.338·35-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36000\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(2.29539\)
Root analytic conductor: \(1.23087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.454301533\)
\(L(\frac12)\) \(\approx\) \(1.454301533\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$ \( 1 + T \)
good7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
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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

−10.03206995033638053454789118892, −9.823375295649081380418590920136, −9.145474696294041331632077665653, −8.761341695645055066667341260395, −8.321722450854969435493752818837, −7.998258853901465580969371367477, −7.28254836118582905874103774127, −6.78923565974098429751523140123, −6.25645156373512622315757491177, −5.32980066926380223263880725199, −4.54010664025825939416315316847, −3.74777133045768309466283974591, −3.32095382498356817710621977137, −2.21911413208596523989656403630, −1.40435653997204247929252733124, 1.40435653997204247929252733124, 2.21911413208596523989656403630, 3.32095382498356817710621977137, 3.74777133045768309466283974591, 4.54010664025825939416315316847, 5.32980066926380223263880725199, 6.25645156373512622315757491177, 6.78923565974098429751523140123, 7.28254836118582905874103774127, 7.998258853901465580969371367477, 8.321722450854969435493752818837, 8.761341695645055066667341260395, 9.145474696294041331632077665653, 9.823375295649081380418590920136, 10.03206995033638053454789118892

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