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Properties

Label 4-3072e2-1.1-c0e2-0-0
Degree $4$
Conductor $9437184$
Sign $1$
Analytic cond. $2.35048$
Root an. cond. $1.23819$
Motivic weight $0$
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  − 9-s − 2·11-s − 2·19-s − 2·43-s + 2·49-s + 2·59-s − 2·67-s + 81-s − 2·83-s − 4·89-s + 2·99-s + 2·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·171-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 9-s − 2·11-s − 2·19-s − 2·43-s + 2·49-s + 2·59-s − 2·67-s + 81-s − 2·83-s − 4·89-s + 2·99-s + 2·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·171-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9437184\)    =    \(2^{20} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2.35048\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9437184,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5171783392\)
\(L(\frac12)\) \(\approx\) \(0.5171783392\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−8.748587718119645416211023497258, −8.615127789385884225835831366804, −8.334956531924677402425154335216, −8.254672618098637174700978003110, −7.48632183372578108001221370051, −7.28313698737298633924444932258, −6.90524386471279579507214698368, −6.37950637268068565718746483443, −5.98098461228239769989367787400, −5.49873122030689284311615634810, −5.47381964105356767147923368354, −4.88667237445933256088071658192, −4.32599727491055933842671294216, −4.17118659043604906272646755827, −3.40131557006202682967208469783, −2.94330913149798377784932212865, −2.60127571082889161222485459384, −2.17775377867540877623160208059, −1.62655482113198672143716053480, −0.41482448358526533172098208651, 0.41482448358526533172098208651, 1.62655482113198672143716053480, 2.17775377867540877623160208059, 2.60127571082889161222485459384, 2.94330913149798377784932212865, 3.40131557006202682967208469783, 4.17118659043604906272646755827, 4.32599727491055933842671294216, 4.88667237445933256088071658192, 5.47381964105356767147923368354, 5.49873122030689284311615634810, 5.98098461228239769989367787400, 6.37950637268068565718746483443, 6.90524386471279579507214698368, 7.28313698737298633924444932258, 7.48632183372578108001221370051, 8.254672618098637174700978003110, 8.334956531924677402425154335216, 8.615127789385884225835831366804, 8.748587718119645416211023497258

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