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Properties

Label 4-1134e2-1.1-c1e2-0-2
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·5-s + 7-s − 8-s − 2·10-s − 4·11-s − 6·13-s + 14-s − 16-s − 4·17-s − 8·19-s − 4·22-s + 8·23-s + 5·25-s − 6·26-s − 2·29-s − 4·34-s − 2·35-s − 20·37-s − 8·38-s + 2·40-s − 6·41-s + 4·43-s + 8·46-s + 5·50-s − 12·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.852·22-s + 1.66·23-s + 25-s − 1.17·26-s − 0.371·29-s − 0.685·34-s − 0.338·35-s − 3.28·37-s − 1.29·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s + 1.17·46-s + 0.707·50-s − 1.64·53-s + 1.07·55-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: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2763208085\)
\(L(\frac12)\) \(\approx\) \(0.2763208085\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_az
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abr
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_az
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.79.a_adb
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_acp
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.ao_dv
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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

−10.49490286720802973267046654617, −9.414940994046001405079471365323, −9.225156556763939278066647661870, −8.589052375645920029969326396141, −8.560338148998537164405704964220, −7.76977622800345579708962766913, −7.68800240946826680593727913074, −6.96370913749017170345196380992, −6.76432306362789556021490776602, −6.42619518965534409040733330419, −5.42631066592115025936896753413, −5.14018480024727090237070710375, −4.97060345755627881930025798668, −4.39605801767808140083202818606, −4.08321840651976725638466863125, −3.25438324421925786745887771407, −2.94731104322690226353970298812, −2.28836752178602409335801169070, −1.73703919101765279046127405963, −0.19229586220280067418553044688, 0.19229586220280067418553044688, 1.73703919101765279046127405963, 2.28836752178602409335801169070, 2.94731104322690226353970298812, 3.25438324421925786745887771407, 4.08321840651976725638466863125, 4.39605801767808140083202818606, 4.97060345755627881930025798668, 5.14018480024727090237070710375, 5.42631066592115025936896753413, 6.42619518965534409040733330419, 6.76432306362789556021490776602, 6.96370913749017170345196380992, 7.68800240946826680593727913074, 7.76977622800345579708962766913, 8.560338148998537164405704964220, 8.589052375645920029969326396141, 9.225156556763939278066647661870, 9.414940994046001405079471365323, 10.49490286720802973267046654617

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