Properties

Label 4-1134e2-1.1-c1e2-0-90
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 4·5-s − 2·7-s + 4·8-s − 8·10-s − 2·11-s − 6·13-s − 4·14-s + 5·16-s − 14·17-s − 2·19-s − 12·20-s − 4·22-s − 2·23-s + 5·25-s − 12·26-s − 6·28-s − 10·29-s + 6·31-s + 6·32-s − 28·34-s + 8·35-s + 4·37-s − 4·38-s − 16·40-s − 12·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.78·5-s − 0.755·7-s + 1.41·8-s − 2.52·10-s − 0.603·11-s − 1.66·13-s − 1.06·14-s + 5/4·16-s − 3.39·17-s − 0.458·19-s − 2.68·20-s − 0.852·22-s − 0.417·23-s + 25-s − 2.35·26-s − 1.13·28-s − 1.85·29-s + 1.07·31-s + 1.06·32-s − 4.80·34-s + 1.35·35-s + 0.657·37-s − 0.648·38-s − 2.52·40-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
11$D_{4}$ \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ae
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.17.o_df
19$D_{4}$ \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_bk
23$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_u
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_ct
31$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_bs
37$D_{4}$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_d
41$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_ec
43$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_da
47$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_dw
53$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_fa
59$D_{4}$ \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_do
61$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_df
67$D_{4}$ \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_ga
71$D_{4}$ \( 1 + 20 T + 230 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_iw
73$D_{4}$ \( 1 - 20 T + 243 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.73.au_jj
79$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_u
83$D_{4}$ \( 1 + 2 T - 76 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_acy
89$D_{4}$ \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_fj
97$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_gg
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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

−9.664074941591599720742791732481, −9.207135692349532420199214553518, −8.567860918077030616389573513678, −8.428341657674312088421045214990, −7.58938048544217216140004445826, −7.56379236408772484960119959418, −6.94693463030764101693840896249, −6.83580903220873777734978824021, −6.17325726335194061442348403962, −5.87373824601335053405090479509, −4.95132725704934400632122717422, −4.79036484471245917661873888650, −4.38862553849927697361249010289, −3.91150210175583802167136505912, −3.61666009803687336262678884369, −2.83636973829316666374829135119, −2.38273707489870961758895142314, −2.00739084460189131272843904705, 0, 0, 2.00739084460189131272843904705, 2.38273707489870961758895142314, 2.83636973829316666374829135119, 3.61666009803687336262678884369, 3.91150210175583802167136505912, 4.38862553849927697361249010289, 4.79036484471245917661873888650, 4.95132725704934400632122717422, 5.87373824601335053405090479509, 6.17325726335194061442348403962, 6.83580903220873777734978824021, 6.94693463030764101693840896249, 7.56379236408772484960119959418, 7.58938048544217216140004445826, 8.428341657674312088421045214990, 8.567860918077030616389573513678, 9.207135692349532420199214553518, 9.664074941591599720742791732481

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