Properties

Label 4-2550e2-1.1-c1e2-0-11
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  − 2·3-s − 4-s − 6·7-s + 3·9-s + 2·12-s + 16-s + 2·17-s + 6·19-s + 12·21-s + 8·23-s − 4·27-s + 6·28-s − 3·36-s − 2·37-s − 2·48-s + 13·49-s − 4·51-s − 12·57-s − 18·63-s − 64-s − 2·68-s − 16·69-s − 16·73-s − 6·76-s + 5·81-s − 12·84-s − 28·89-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 2.26·7-s + 9-s + 0.577·12-s + 1/4·16-s + 0.485·17-s + 1.37·19-s + 2.61·21-s + 1.66·23-s − 0.769·27-s + 1.13·28-s − 1/2·36-s − 0.328·37-s − 0.288·48-s + 13/7·49-s − 0.560·51-s − 1.58·57-s − 2.26·63-s − 1/8·64-s − 0.242·68-s − 1.92·69-s − 1.87·73-s − 0.688·76-s + 5/9·81-s − 1.30·84-s − 2.96·89-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

\(L(1)\) \(\approx\) \(0.5803039027\)
\(L(\frac12)\) \(\approx\) \(0.5803039027\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−9.297587577709083917083520511908, −8.892756424008541898077309186296, −8.369503059379697741441882855301, −7.899688063322456384551333272838, −7.25098975320666574432081225379, −7.05295906452709751095347246789, −6.94918011296561428514612802274, −6.16151460671076838923738610434, −6.13985253607966871768912992837, −5.57909235573155867562641195890, −5.34093346229863907654170012931, −4.75396057484050811117553628082, −4.50113188680456791139566161954, −3.62137692421882928933632210798, −3.58153650663541540634587752017, −2.94571726199328700116265036655, −2.72678334079770649626699172115, −1.56788723956081712455744298736, −0.994487031384995753286663556856, −0.34550019099306317638419760254, 0.34550019099306317638419760254, 0.994487031384995753286663556856, 1.56788723956081712455744298736, 2.72678334079770649626699172115, 2.94571726199328700116265036655, 3.58153650663541540634587752017, 3.62137692421882928933632210798, 4.50113188680456791139566161954, 4.75396057484050811117553628082, 5.34093346229863907654170012931, 5.57909235573155867562641195890, 6.13985253607966871768912992837, 6.16151460671076838923738610434, 6.94918011296561428514612802274, 7.05295906452709751095347246789, 7.25098975320666574432081225379, 7.899688063322456384551333272838, 8.369503059379697741441882855301, 8.892756424008541898077309186296, 9.297587577709083917083520511908

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