↑

Properties

Label 4-7104e2-1.1-c1e2-0-0
Degree $4$
Conductor $50466816$
Sign $1$
Analytic cond. $3217.80$
Root an. cond. $7.53164$
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  − 2·3-s − 2·5-s − 8·7-s + 3·9-s + 4·11-s + 4·15-s − 6·17-s + 16·21-s − 2·23-s − 2·25-s − 4·27-s + 6·29-s − 12·31-s − 8·33-s + 16·35-s − 2·37-s − 4·41-s + 4·43-s − 6·45-s − 16·47-s + 34·49-s + 12·51-s − 8·53-s − 8·55-s + 2·59-s + 12·61-s − 24·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 3.02·7-s + 9-s + 1.20·11-s + 1.03·15-s − 1.45·17-s + 3.49·21-s − 0.417·23-s − 2/5·25-s − 0.769·27-s + 1.11·29-s − 2.15·31-s − 1.39·33-s + 2.70·35-s − 0.328·37-s − 0.624·41-s + 0.609·43-s − 0.894·45-s − 2.33·47-s + 34/7·49-s + 1.68·51-s − 1.09·53-s − 1.07·55-s + 0.260·59-s + 1.53·61-s − 3.02·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50466816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50466816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50466816\)    =    \(2^{12} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(3217.80\)
Root analytic conductor: \(7.53164\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50466816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3090871381\)
\(L(\frac12)\) \(\approx\) \(0.3090871381\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_w
31$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_da
41$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_g
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_cs
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_ag
61$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_da
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_da
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fm
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.83.u_jm
89$D_{4}$ \( 1 - 22 T + 254 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.89.aw_ju
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.97.a_o
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

−7.987701642509487313713570245448, −7.64103397713817802821444478101, −7.18897097665684387528937535789, −6.87661178761912228051069824149, −6.55921701220043058458824166469, −6.48200796208001605845904445843, −6.24565089420628760623605522956, −5.80004311699175008769353452382, −5.34731766534395716547651356651, −4.97527985474830191533505902412, −4.36505029636170049667188721752, −4.17728175990688111935683440489, −3.61006636559419061203383258451, −3.54246407423164886663315029341, −3.22248312428592357133900169049, −2.54805955988543576136377641519, −2.01328074946836453628233535348, −1.46936854159902414952259544481, −0.54096091468856009548009605746, −0.27681968000834722022311559444, 0.27681968000834722022311559444, 0.54096091468856009548009605746, 1.46936854159902414952259544481, 2.01328074946836453628233535348, 2.54805955988543576136377641519, 3.22248312428592357133900169049, 3.54246407423164886663315029341, 3.61006636559419061203383258451, 4.17728175990688111935683440489, 4.36505029636170049667188721752, 4.97527985474830191533505902412, 5.34731766534395716547651356651, 5.80004311699175008769353452382, 6.24565089420628760623605522956, 6.48200796208001605845904445843, 6.55921701220043058458824166469, 6.87661178761912228051069824149, 7.18897097665684387528937535789, 7.64103397713817802821444478101, 7.987701642509487313713570245448

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