Properties

Label 4-5775e2-1.1-c1e2-0-2
Degree $4$
Conductor $33350625$
Sign $1$
Analytic cond. $2126.46$
Root an. cond. $6.79070$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 2·4-s + 2·6-s − 2·7-s + 5·8-s + 3·9-s − 2·11-s + 4·12-s − 2·13-s − 2·14-s + 5·16-s − 6·17-s + 3·18-s − 4·19-s − 4·21-s − 2·22-s + 2·23-s + 10·24-s − 2·26-s + 4·27-s − 4·28-s + 2·29-s − 2·31-s + 10·32-s − 4·33-s − 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 4-s + 0.816·6-s − 0.755·7-s + 1.76·8-s + 9-s − 0.603·11-s + 1.15·12-s − 0.554·13-s − 0.534·14-s + 5/4·16-s − 1.45·17-s + 0.707·18-s − 0.917·19-s − 0.872·21-s − 0.426·22-s + 0.417·23-s + 2.04·24-s − 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.371·29-s − 0.359·31-s + 1.76·32-s − 0.696·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33350625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33350625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33350625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2126.46\)
Root analytic conductor: \(6.79070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33350625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.283658334\)
\(L(\frac12)\) \(\approx\) \(6.283658334\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 129 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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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

−8.474488482482199247999781927299, −7.70935676772730314395657631840, −7.52451721303055805792350413210, −7.28248503101945853814761201446, −6.89888355717479577768646063098, −6.46734210383121629068220646273, −6.42760518470388660025804293608, −5.85118580359455104610255846593, −5.11963274563894511131504750552, −5.11266616425640761138173681536, −4.42901734059988168403680239493, −4.40340461306776061556222345749, −3.71190676461332059675438448282, −3.57215432253729357472842397096, −2.95425203736711203862040693690, −2.43378164829561393571288384002, −2.36800610064009426129857228553, −1.92154049140804054072943268368, −1.34711177873983303269355542394, −0.48105070852092504498161870187, 0.48105070852092504498161870187, 1.34711177873983303269355542394, 1.92154049140804054072943268368, 2.36800610064009426129857228553, 2.43378164829561393571288384002, 2.95425203736711203862040693690, 3.57215432253729357472842397096, 3.71190676461332059675438448282, 4.40340461306776061556222345749, 4.42901734059988168403680239493, 5.11266616425640761138173681536, 5.11963274563894511131504750552, 5.85118580359455104610255846593, 6.42760518470388660025804293608, 6.46734210383121629068220646273, 6.89888355717479577768646063098, 7.28248503101945853814761201446, 7.52451721303055805792350413210, 7.70935676772730314395657631840, 8.474488482482199247999781927299

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