Properties

Label 4-456300-1.1-c1e2-0-5
Degree $4$
Conductor $456300$
Sign $-1$
Analytic cond. $29.0940$
Root an. cond. $2.32247$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 9-s − 12-s − 2·13-s + 16-s + 25-s − 27-s + 36-s − 12·37-s + 2·39-s − 8·43-s − 48-s − 14·49-s − 2·52-s + 12·61-s + 64-s + 8·67-s − 4·73-s − 75-s + 81-s + 28·97-s + 100-s + 24·103-s − 108-s − 28·109-s + 12·111-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 1/5·25-s − 0.192·27-s + 1/6·36-s − 1.97·37-s + 0.320·39-s − 1.21·43-s − 0.144·48-s − 2·49-s − 0.277·52-s + 1.53·61-s + 1/8·64-s + 0.977·67-s − 0.468·73-s − 0.115·75-s + 1/9·81-s + 2.84·97-s + 1/10·100-s + 2.36·103-s − 0.0962·108-s − 2.68·109-s + 1.13·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(456300\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(29.0940\)
Root analytic conductor: \(2.32247\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 456300,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−8.072251512020700899711976441689, −8.062771471643955273434189645297, −7.13489552716111364035097044906, −7.11791279895358735884585299226, −6.36199631841087100215380294304, −6.27139115246731968799115000690, −5.39759482678390501023193939738, −5.02996251545709838502394177698, −4.78086243570270981436649484362, −3.75949148792159340735853491565, −3.52270970074791721181391566206, −2.67714551530238471219197858825, −2.01716871449082204952154497689, −1.30131976065987787395283637597, 0, 1.30131976065987787395283637597, 2.01716871449082204952154497689, 2.67714551530238471219197858825, 3.52270970074791721181391566206, 3.75949148792159340735853491565, 4.78086243570270981436649484362, 5.02996251545709838502394177698, 5.39759482678390501023193939738, 6.27139115246731968799115000690, 6.36199631841087100215380294304, 7.11791279895358735884585299226, 7.13489552716111364035097044906, 8.062771471643955273434189645297, 8.072251512020700899711976441689

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