Properties

Label 4-42e4-1.1-c1e2-0-11
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·13-s + 8·19-s + 5·25-s − 4·31-s + 10·37-s + 16·43-s + 14·61-s + 16·67-s − 10·73-s + 4·79-s − 28·97-s + 20·103-s − 2·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.10·13-s + 1.83·19-s + 25-s − 0.718·31-s + 1.64·37-s + 2.43·43-s + 1.79·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s − 2.84·97-s + 1.97·103-s − 0.191·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.527861701\)
\(L(\frac12)\) \(\approx\) \(2.527861701\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−9.530044344664672823687370209582, −9.361189171956335614796828198990, −8.516798118005930908986577574619, −8.472291674775557307128830219238, −7.86147694135727985556909256989, −7.39745551303209634332365244822, −7.07345698940820125811355153222, −7.07050729325715355674057844797, −6.15921684507740690464974393958, −5.88020959733861392847866171818, −5.36215989589383442813981310633, −5.13500414511092058323493125901, −4.45113063728847987930472033292, −4.26571316277168790413314224553, −3.46801658604687274419207993817, −3.14774433695310256136608703272, −2.45455288832451320950951149337, −2.22379300296538333079826554310, −1.14673234245407380756870781757, −0.69335406972705442880699528951, 0.69335406972705442880699528951, 1.14673234245407380756870781757, 2.22379300296538333079826554310, 2.45455288832451320950951149337, 3.14774433695310256136608703272, 3.46801658604687274419207993817, 4.26571316277168790413314224553, 4.45113063728847987930472033292, 5.13500414511092058323493125901, 5.36215989589383442813981310633, 5.88020959733861392847866171818, 6.15921684507740690464974393958, 7.07050729325715355674057844797, 7.07345698940820125811355153222, 7.39745551303209634332365244822, 7.86147694135727985556909256989, 8.472291674775557307128830219238, 8.516798118005930908986577574619, 9.361189171956335614796828198990, 9.530044344664672823687370209582

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