Properties

Label 4-4026e2-1.1-c1e2-0-1
Degree $4$
Conductor $16208676$
Sign $1$
Analytic cond. $1033.47$
Root an. cond. $5.66990$
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  + 2·2-s − 2·3-s + 3·4-s − 3·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 6·10-s + 2·11-s − 6·12-s + 5·13-s − 4·14-s + 6·15-s + 5·16-s + 2·17-s + 6·18-s − 8·19-s − 9·20-s + 4·21-s + 4·22-s − 8·24-s + 25-s + 10·26-s − 4·27-s − 6·28-s + 5·29-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.89·10-s + 0.603·11-s − 1.73·12-s + 1.38·13-s − 1.06·14-s + 1.54·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 1.83·19-s − 2.01·20-s + 0.872·21-s + 0.852·22-s − 1.63·24-s + 1/5·25-s + 1.96·26-s − 0.769·27-s − 1.13·28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16208676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16208676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16208676\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 61^{2}\)
Sign: $1$
Analytic conductor: \(1033.47\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16208676,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.066148227\)
\(L(\frac12)\) \(\approx\) \(4.066148227\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
61$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 19 T + 168 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−8.569760112048533711495903405879, −8.186271738488024773627791442517, −7.61352489752610147560192472069, −7.39399973736778786757092821524, −7.09878355214631797302430913557, −6.48162928746938984919413204371, −6.29370509111723728189056254365, −6.11124140920893868339343789696, −5.78392269857247572995609114421, −5.29442640203350541182991335888, −4.61063074669619040841600316742, −4.53681080622085118219248382046, −3.99439497616293940872971858151, −3.90278781533997207699035937457, −3.32799247573596868477010843854, −3.13267518387255519220683094336, −2.16369845345674874245726643580, −1.94583125093181597011122366542, −0.789791873803405629986404740523, −0.71804524588869491190238587920, 0.71804524588869491190238587920, 0.789791873803405629986404740523, 1.94583125093181597011122366542, 2.16369845345674874245726643580, 3.13267518387255519220683094336, 3.32799247573596868477010843854, 3.90278781533997207699035937457, 3.99439497616293940872971858151, 4.53681080622085118219248382046, 4.61063074669619040841600316742, 5.29442640203350541182991335888, 5.78392269857247572995609114421, 6.11124140920893868339343789696, 6.29370509111723728189056254365, 6.48162928746938984919413204371, 7.09878355214631797302430913557, 7.39399973736778786757092821524, 7.61352489752610147560192472069, 8.186271738488024773627791442517, 8.569760112048533711495903405879

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