Properties

Label 4-1344e2-1.1-c1e2-0-18
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
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  − 3-s − 2·5-s + 5·7-s + 2·11-s − 2·13-s + 2·15-s − 19-s − 5·21-s + 5·25-s + 27-s − 8·29-s + 9·31-s − 2·33-s − 10·35-s + 3·37-s + 2·39-s − 20·41-s + 10·43-s − 6·47-s + 18·49-s + 12·53-s − 4·55-s + 57-s + 12·59-s + 10·61-s + 4·65-s + 5·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.88·7-s + 0.603·11-s − 0.554·13-s + 0.516·15-s − 0.229·19-s − 1.09·21-s + 25-s + 0.192·27-s − 1.48·29-s + 1.61·31-s − 0.348·33-s − 1.69·35-s + 0.493·37-s + 0.320·39-s − 3.12·41-s + 1.52·43-s − 0.875·47-s + 18/7·49-s + 1.64·53-s − 0.539·55-s + 0.132·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 0.610·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.843303192\)
\(L(\frac12)\) \(\approx\) \(1.843303192\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.879539926584021565820913610726, −9.477191695333320814957310857588, −8.755643028478969921157161689300, −8.481396352938125284172544178984, −8.254018778702492115777506924569, −7.940306457245540168972898399719, −7.20585986520427073943311186500, −7.08163165525021728070270361867, −6.73774166015014687828169200840, −5.97964260102715271669081659313, −5.55468133091786244544464604440, −5.13653719688575043156734045298, −4.66765624712245348126363793793, −4.50502807992162798762464773909, −3.68648320270898682946041825387, −3.59685979068361580506779266444, −2.45317377498539685267781440697, −2.13583310164504635671573446885, −1.28321134617165548338199209428, −0.64932361999403692223442091207, 0.64932361999403692223442091207, 1.28321134617165548338199209428, 2.13583310164504635671573446885, 2.45317377498539685267781440697, 3.59685979068361580506779266444, 3.68648320270898682946041825387, 4.50502807992162798762464773909, 4.66765624712245348126363793793, 5.13653719688575043156734045298, 5.55468133091786244544464604440, 5.97964260102715271669081659313, 6.73774166015014687828169200840, 7.08163165525021728070270361867, 7.20585986520427073943311186500, 7.940306457245540168972898399719, 8.254018778702492115777506924569, 8.481396352938125284172544178984, 8.755643028478969921157161689300, 9.477191695333320814957310857588, 9.879539926584021565820913610726

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