Properties

Label 4-2300e2-1.1-c1e2-0-0
Degree $4$
Conductor $5290000$
Sign $1$
Analytic cond. $337.294$
Root an. cond. $4.28550$
Motivic weight $1$
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  + 5·9-s − 12·11-s − 4·19-s − 18·29-s + 10·31-s − 18·41-s − 2·49-s + 4·61-s − 6·71-s + 20·79-s + 16·81-s − 60·99-s − 12·101-s − 40·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s − 20·171-s + 173-s + ⋯
L(s)  = 1  + 5/3·9-s − 3.61·11-s − 0.917·19-s − 3.34·29-s + 1.79·31-s − 2.81·41-s − 2/7·49-s + 0.512·61-s − 0.712·71-s + 2.25·79-s + 16/9·81-s − 6.03·99-s − 1.19·101-s − 3.83·109-s + 7.81·121-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 + 1.92·169-s − 1.52·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5290000\)    =    \(2^{4} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(337.294\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2506849310\)
\(L(\frac12)\) \(\approx\) \(0.2506849310\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.460606345505189938854885297554, −8.716885702611286570852312680671, −8.176555094799177581718049673505, −8.019193622759623204402769801509, −7.79196115165720721342594144563, −7.25909361378451608186088683608, −7.08481640846764953556820124622, −6.49724703103641088089339742306, −6.13242561854123591160145229221, −5.32333015488759902850864674777, −5.31770153608145184262237174947, −4.94813984113802290231143898505, −4.51362422315349376901523685725, −3.72345083160389031744303056907, −3.70542405516851421884586547257, −2.73894347692831500873402811056, −2.51207310967288742439514606337, −1.93780567880003502341009651791, −1.42286366840267562640283688579, −0.16339321070087693387632007168, 0.16339321070087693387632007168, 1.42286366840267562640283688579, 1.93780567880003502341009651791, 2.51207310967288742439514606337, 2.73894347692831500873402811056, 3.70542405516851421884586547257, 3.72345083160389031744303056907, 4.51362422315349376901523685725, 4.94813984113802290231143898505, 5.31770153608145184262237174947, 5.32333015488759902850864674777, 6.13242561854123591160145229221, 6.49724703103641088089339742306, 7.08481640846764953556820124622, 7.25909361378451608186088683608, 7.79196115165720721342594144563, 8.019193622759623204402769801509, 8.176555094799177581718049673505, 8.716885702611286570852312680671, 9.460606345505189938854885297554

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