↑

Properties

Label 4-1776e2-1.1-c1e2-0-18
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $201.112$
Root an. cond. $3.76582$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 7-s + 4·11-s − 3·13-s + 4·15-s − 8·17-s − 8·19-s − 21-s + 5·25-s + 27-s − 20·29-s + 14·31-s − 4·33-s − 4·35-s + 10·37-s + 3·39-s + 2·41-s − 10·43-s + 12·47-s + 7·49-s + 8·51-s − 6·53-s − 16·55-s + 8·57-s − 12·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 0.377·7-s + 1.20·11-s − 0.832·13-s + 1.03·15-s − 1.94·17-s − 1.83·19-s − 0.218·21-s + 25-s + 0.192·27-s − 3.71·29-s + 2.51·31-s − 0.696·33-s − 0.676·35-s + 1.64·37-s + 0.480·39-s + 0.312·41-s − 1.52·43-s + 1.75·47-s + 49-s + 1.12·51-s − 0.824·53-s − 2.15·55-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(201.112\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3154176,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
37$C_2$ \( 1 - 10 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_ae
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bv
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.i_bt
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.31.ao_eh
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.43.k_eh
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.67.aj_o
71$C_2^2$ \( 1 - 10 T + 29 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_bd
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.79.j_c
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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

−9.254954390727516434006062771942, −8.777498966348514908820641154364, −8.118898558602027675216663693374, −8.039462635349132737902234764260, −7.36763639073063772048789327483, −7.28109512434173496778651915468, −6.71955138927057100818498158492, −6.24761086752910059336088168915, −6.07934058666188389188393230863, −5.43922232891893558136014620058, −4.66520700221105403108783765834, −4.50715538721942426322936270007, −4.05130963147098062814998672608, −3.99439711479337234192795878666, −3.23740954963212822670848682616, −2.37118975271917916129188110272, −2.12978183694363581366358963616, −1.18964995933106277204164187028, 0, 0, 1.18964995933106277204164187028, 2.12978183694363581366358963616, 2.37118975271917916129188110272, 3.23740954963212822670848682616, 3.99439711479337234192795878666, 4.05130963147098062814998672608, 4.50715538721942426322936270007, 4.66520700221105403108783765834, 5.43922232891893558136014620058, 6.07934058666188389188393230863, 6.24761086752910059336088168915, 6.71955138927057100818498158492, 7.28109512434173496778651915468, 7.36763639073063772048789327483, 8.039462635349132737902234764260, 8.118898558602027675216663693374, 8.777498966348514908820641154364, 9.254954390727516434006062771942

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