Properties

Label 4-36000-1.1-c1e2-0-7
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

\(L(1)\) \(\approx\) \(2.886019006\)
\(L(\frac12)\) \(\approx\) \(2.886019006\)
\(L(\frac{3}{2})\) 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$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
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

−10.26550284089543110185358487309, −9.939532220906082292938318921999, −9.373845629022664004351492583789, −8.514395113889280353725630614232, −8.225357992577434630349037419547, −7.935900977294712502169202954837, −7.16693466415474646896878501510, −6.68235953578835017677243281762, −5.72538334962941231954188927789, −5.25698516118922516203421497112, −4.83974596893253254120988786028, −3.89888547911762517181730953593, −3.15765301812739655649310440328, −2.50047338717890012999824342531, −1.86710543653544131760149222635, 1.86710543653544131760149222635, 2.50047338717890012999824342531, 3.15765301812739655649310440328, 3.89888547911762517181730953593, 4.83974596893253254120988786028, 5.25698516118922516203421497112, 5.72538334962941231954188927789, 6.68235953578835017677243281762, 7.16693466415474646896878501510, 7.935900977294712502169202954837, 8.225357992577434630349037419547, 8.514395113889280353725630614232, 9.373845629022664004351492583789, 9.939532220906082292938318921999, 10.26550284089543110185358487309

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