Properties

Label 4-300e2-1.1-c2e2-0-3
Degree $4$
Conductor $90000$
Sign $1$
Analytic cond. $66.8209$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·9-s − 52·19-s − 92·31-s + 94·49-s + 148·61-s + 284·79-s + 81·81-s + 428·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + 468·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 9-s − 2.73·19-s − 2.96·31-s + 1.91·49-s + 2.42·61-s + 3.59·79-s + 81-s + 3.92·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + 2.73·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(66.8209\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.090465667\)
\(L(\frac12)\) \(\approx\) \(1.090465667\)
\(L(2)\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 94 T^{2} + p^{4} T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2^2$ \( 1 + 146 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2062 T^{2} + p^{4} T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 - 3214 T^{2} + p^{4} T^{4} \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 5906 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 8542 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 - 18814 T^{2} + p^{4} T^{4} \)
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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

−11.65894457312569996036557439796, −11.05554593106336849182783927498, −11.02248816653733503534775540612, −10.44235121939280235955224392967, −9.997613870902732190259176458328, −9.106530733035266101682380655753, −9.017331377980182466971970882402, −8.469124244425378967815587703750, −8.084862373590894508443976560871, −7.32157117934016320868338868758, −6.94767676644916616494534931885, −6.21991501236242035894940905654, −5.88583030440727972690620875082, −5.27202927199347490164345987491, −4.66485083460652070047981802162, −3.80122920765454181374172985339, −3.56643886117874732401939577164, −2.21309151752648367900254771234, −2.18127135462061597737397001789, −0.48535084706982882982774333539, 0.48535084706982882982774333539, 2.18127135462061597737397001789, 2.21309151752648367900254771234, 3.56643886117874732401939577164, 3.80122920765454181374172985339, 4.66485083460652070047981802162, 5.27202927199347490164345987491, 5.88583030440727972690620875082, 6.21991501236242035894940905654, 6.94767676644916616494534931885, 7.32157117934016320868338868758, 8.084862373590894508443976560871, 8.469124244425378967815587703750, 9.017331377980182466971970882402, 9.106530733035266101682380655753, 9.997613870902732190259176458328, 10.44235121939280235955224392967, 11.02248816653733503534775540612, 11.05554593106336849182783927498, 11.65894457312569996036557439796

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