Properties

Degree $4$
Conductor $345744$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 2·11-s + 8·13-s + 2·15-s + 6·17-s + 8·19-s + 6·23-s + 5·25-s − 27-s − 20·29-s + 4·31-s − 2·33-s − 6·37-s + 8·39-s + 12·41-s + 8·43-s + 8·47-s + 6·51-s − 2·53-s − 4·55-s + 8·57-s − 4·59-s − 8·61-s + 16·65-s + 8·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.603·11-s + 2.21·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s + 1.25·23-s + 25-s − 0.192·27-s − 3.71·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 1.28·39-s + 1.87·41-s + 1.21·43-s + 1.16·47-s + 0.840·51-s − 0.274·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s − 1.02·61-s + 1.98·65-s + 0.977·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{588} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 345744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.27866\)
\(L(\frac12)\) \(\approx\) \(3.27866\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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

−10.73965459330473160920261018699, −10.72778583873254232817321505124, −9.731319955720249993534157848566, −9.688946276586426160804765139668, −9.142962783455452026301481432854, −8.835877260878964498105470781262, −8.412426757454182326040029176675, −7.63984826694279960957230548952, −7.45629129092727535273265018336, −7.14265581430412680312797508121, −6.11892232742314159672942746422, −5.85860392646274529244502887464, −5.47156165595483659599096975733, −5.18891765822095017483420416119, −4.01716907375447724638111147930, −3.78106439838390656310252233410, −2.91728336039430706781360995339, −2.82157880809136794235737109534, −1.48454647706044124862202797041, −1.23553727931513015429673319306, 1.23553727931513015429673319306, 1.48454647706044124862202797041, 2.82157880809136794235737109534, 2.91728336039430706781360995339, 3.78106439838390656310252233410, 4.01716907375447724638111147930, 5.18891765822095017483420416119, 5.47156165595483659599096975733, 5.85860392646274529244502887464, 6.11892232742314159672942746422, 7.14265581430412680312797508121, 7.45629129092727535273265018336, 7.63984826694279960957230548952, 8.412426757454182326040029176675, 8.835877260878964498105470781262, 9.142962783455452026301481432854, 9.688946276586426160804765139668, 9.731319955720249993534157848566, 10.72778583873254232817321505124, 10.73965459330473160920261018699

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