Properties

Label 4-2550e2-1.1-c1e2-0-6
Degree $4$
Conductor $6502500$
Sign $1$
Analytic cond. $414.605$
Root an. cond. $4.51241$
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  − 4-s − 9-s − 10·11-s + 16-s + 2·19-s + 8·29-s − 2·31-s + 36-s − 20·41-s + 10·44-s + 5·49-s + 16·59-s − 28·61-s − 64-s + 28·71-s − 2·76-s + 10·79-s + 81-s + 4·89-s + 10·99-s + 18·101-s − 2·109-s − 8·116-s + 53·121-s + 2·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 3.01·11-s + 1/4·16-s + 0.458·19-s + 1.48·29-s − 0.359·31-s + 1/6·36-s − 3.12·41-s + 1.50·44-s + 5/7·49-s + 2.08·59-s − 3.58·61-s − 1/8·64-s + 3.32·71-s − 0.229·76-s + 1.12·79-s + 1/9·81-s + 0.423·89-s + 1.00·99-s + 1.79·101-s − 0.191·109-s − 0.742·116-s + 4.81·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6502500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(414.605\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6502500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5356257372\)
\(L(\frac12)\) \(\approx\) \(0.5356257372\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.872170636558553188906265883403, −8.758228940635798909519084726514, −8.122621505622242744883524592800, −8.120594977203527749528302513906, −7.65258197608749727134915894114, −7.36445461508955734346350956935, −6.75373804742397315424181690602, −6.43718923267881036432171038090, −5.90108256409752210919316025928, −5.36862531987375799769828732567, −5.18741852439797143034511018518, −4.84312607809245941682075205108, −4.62484371700703478911762445284, −3.58569093515897605389985210824, −3.51171618457979135614276262987, −2.87514947926548939877522874911, −2.44386134705013953532269830688, −2.06017842999979627194415357239, −1.10343457171104092569875563858, −0.26533985847052469959919698792, 0.26533985847052469959919698792, 1.10343457171104092569875563858, 2.06017842999979627194415357239, 2.44386134705013953532269830688, 2.87514947926548939877522874911, 3.51171618457979135614276262987, 3.58569093515897605389985210824, 4.62484371700703478911762445284, 4.84312607809245941682075205108, 5.18741852439797143034511018518, 5.36862531987375799769828732567, 5.90108256409752210919316025928, 6.43718923267881036432171038090, 6.75373804742397315424181690602, 7.36445461508955734346350956935, 7.65258197608749727134915894114, 8.120594977203527749528302513906, 8.122621505622242744883524592800, 8.758228940635798909519084726514, 8.872170636558553188906265883403

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