Properties

Label 4-901404-1.1-c1e2-0-4
Degree $4$
Conductor $901404$
Sign $-1$
Analytic cond. $57.4743$
Root an. cond. $2.75339$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 7-s − 3·9-s − 6·13-s + 16-s + 8·25-s + 28-s − 6·31-s + 3·36-s + 8·37-s + 8·43-s + 49-s + 6·52-s − 12·61-s + 3·63-s − 64-s + 4·67-s + 17·73-s − 2·79-s + 9·81-s + 6·91-s − 8·100-s − 18·103-s − 28·109-s − 112-s + 18·117-s + 4·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.377·7-s − 9-s − 1.66·13-s + 1/4·16-s + 8/5·25-s + 0.188·28-s − 1.07·31-s + 1/2·36-s + 1.31·37-s + 1.21·43-s + 1/7·49-s + 0.832·52-s − 1.53·61-s + 0.377·63-s − 1/8·64-s + 0.488·67-s + 1.98·73-s − 0.225·79-s + 81-s + 0.628·91-s − 4/5·100-s − 1.77·103-s − 2.68·109-s − 0.0944·112-s + 1.66·117-s + 4/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(901404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73\)
Sign: $-1$
Analytic conductor: \(57.4743\)
Root analytic conductor: \(2.75339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{901404} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( 1 + T \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 16 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.054164276153657025216058182678, −7.49295013671548862296143041230, −7.13641414814088792697999514613, −6.66959596408741876407724833667, −6.12723577952151804587764415774, −5.66161128027398707240359691807, −5.17871363454874673458612847932, −4.85937608862981009111570410509, −4.28519418594894667147101211517, −3.74947567953645638022669054009, −2.96297744721935200465180770406, −2.74154791529486118242986057438, −2.07424052785995222127260664669, −0.920448659832283933640261317476, 0, 0.920448659832283933640261317476, 2.07424052785995222127260664669, 2.74154791529486118242986057438, 2.96297744721935200465180770406, 3.74947567953645638022669054009, 4.28519418594894667147101211517, 4.85937608862981009111570410509, 5.17871363454874673458612847932, 5.66161128027398707240359691807, 6.12723577952151804587764415774, 6.66959596408741876407724833667, 7.13641414814088792697999514613, 7.49295013671548862296143041230, 8.054164276153657025216058182678

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