Properties

Label 4-614e2-1.1-c1e2-0-0
Degree $4$
Conductor $376996$
Sign $1$
Analytic cond. $24.0375$
Root an. cond. $2.21423$
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  + 2·2-s − 6·3-s + 3·4-s − 4·5-s − 12·6-s + 4·8-s + 21·9-s − 8·10-s − 3·11-s − 18·12-s + 6·13-s + 24·15-s + 5·16-s − 14·17-s + 42·18-s − 8·19-s − 12·20-s − 6·22-s − 4·23-s − 24·24-s + 5·25-s + 12·26-s − 54·27-s + 6·29-s + 48·30-s − 10·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 3.46·3-s + 3/2·4-s − 1.78·5-s − 4.89·6-s + 1.41·8-s + 7·9-s − 2.52·10-s − 0.904·11-s − 5.19·12-s + 1.66·13-s + 6.19·15-s + 5/4·16-s − 3.39·17-s + 9.89·18-s − 1.83·19-s − 2.68·20-s − 1.27·22-s − 0.834·23-s − 4.89·24-s + 25-s + 2.35·26-s − 10.3·27-s + 1.11·29-s + 8.76·30-s − 1.79·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 376996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(376996\)    =    \(2^{2} \cdot 307^{2}\)
Sign: $1$
Analytic conductor: \(24.0375\)
Root analytic conductor: \(2.21423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 376996,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
307$C_2$ \( 1 + 16 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \) 2.3.g_p
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.17.o_df
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_cr
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ab
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.41.ah_i
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) 2.59.b_acg
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) 2.67.b_aco
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 + 8 T - 19 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_at
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_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

−10.70287213437937646080513526332, −10.65196960892795150609847448475, −10.14798684550299299211691660959, −9.102737053950518779678900393431, −8.381265299917251239658185324954, −8.147635050496111421369184233439, −7.17124696425949590876184801465, −7.04482225332895130230973359205, −6.52422749793746797491439416523, −6.22055861452798134902564778165, −5.89629396689026845443342249930, −5.38865811648674766328963408122, −4.68279331878965062889719623598, −4.45661793875446283633316228410, −4.08333048207570170320155636318, −3.88729562333261088718551048360, −2.47155195514422608380179301057, −1.52358858231315068307710758304, 0, 0, 1.52358858231315068307710758304, 2.47155195514422608380179301057, 3.88729562333261088718551048360, 4.08333048207570170320155636318, 4.45661793875446283633316228410, 4.68279331878965062889719623598, 5.38865811648674766328963408122, 5.89629396689026845443342249930, 6.22055861452798134902564778165, 6.52422749793746797491439416523, 7.04482225332895130230973359205, 7.17124696425949590876184801465, 8.147635050496111421369184233439, 8.381265299917251239658185324954, 9.102737053950518779678900393431, 10.14798684550299299211691660959, 10.65196960892795150609847448475, 10.70287213437937646080513526332

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