Properties

Degree $4$
Conductor $23040000$
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  − 9-s − 8·19-s − 12·29-s + 16·31-s − 12·41-s − 2·49-s + 20·61-s − 16·79-s + 81-s − 36·89-s − 36·101-s − 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 8·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.83·19-s − 2.22·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 2.56·61-s − 1.80·79-s + 1/9·81-s − 3.81·89-s − 3.58·101-s − 1.91·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.611·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23040000\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2683744567\)
\(L(\frac12)\) \(\approx\) \(0.2683744567\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.455233135705387212861052314482, −8.158297965893395875536536566238, −8.029451383413718376665135192763, −7.19546948894517705814261930376, −7.00973490429193110249063852313, −6.56133756697981748068188510870, −6.53632454707921671253133712783, −5.74118921287138040355002067898, −5.65708101710557455717847389203, −5.25132457113131706421315713230, −4.70127456620026194380752292530, −4.33627945846281192038255049054, −3.94566661095528265927170750141, −3.70510808288950872780027081159, −2.97933740423861091418010559601, −2.59785736426635415508276587317, −2.31470118259016022338550526527, −1.56798595818064913692494655373, −1.24523399701690396111495928999, −0.14036159239926709240687008141, 0.14036159239926709240687008141, 1.24523399701690396111495928999, 1.56798595818064913692494655373, 2.31470118259016022338550526527, 2.59785736426635415508276587317, 2.97933740423861091418010559601, 3.70510808288950872780027081159, 3.94566661095528265927170750141, 4.33627945846281192038255049054, 4.70127456620026194380752292530, 5.25132457113131706421315713230, 5.65708101710557455717847389203, 5.74118921287138040355002067898, 6.53632454707921671253133712783, 6.56133756697981748068188510870, 7.00973490429193110249063852313, 7.19546948894517705814261930376, 8.029451383413718376665135192763, 8.158297965893395875536536566238, 8.455233135705387212861052314482

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