Properties

Label 4-1100e2-1.1-c1e2-0-4
Degree $4$
Conductor $1210000$
Sign $1$
Analytic cond. $77.1506$
Root an. cond. $2.96370$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5·7-s − 2·11-s + 2·13-s + 3·17-s + 4·19-s + 5·21-s + 3·23-s + 2·27-s + 9·29-s − 8·31-s − 2·33-s − 4·37-s + 2·39-s − 12·41-s + 20·43-s + 12·47-s + 10·49-s + 3·51-s − 9·53-s + 4·57-s − 12·59-s + 13·61-s + 8·67-s + 3·69-s + 6·71-s + 11·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.88·7-s − 0.603·11-s + 0.554·13-s + 0.727·17-s + 0.917·19-s + 1.09·21-s + 0.625·23-s + 0.384·27-s + 1.67·29-s − 1.43·31-s − 0.348·33-s − 0.657·37-s + 0.320·39-s − 1.87·41-s + 3.04·43-s + 1.75·47-s + 10/7·49-s + 0.420·51-s − 1.23·53-s + 0.529·57-s − 1.56·59-s + 1.66·61-s + 0.977·67-s + 0.361·69-s + 0.712·71-s + 1.28·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1210000\)    =    \(2^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(77.1506\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1210000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.922097218\)
\(L(\frac12)\) \(\approx\) \(3.922097218\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_b
7$D_{4}$ \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.7.af_p
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$D_{4}$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_bf
19$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_v
23$D_{4}$ \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_br
29$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_cv
31$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cf
37$D_{4}$ \( 1 + 4 T + 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cf
41$D_{4}$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dt
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$D_{4}$ \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_ef
53$D_{4}$ \( 1 + 9 T + 79 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_db
59$D_{4}$ \( 1 + 12 T + 133 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fd
61$D_{4}$ \( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.61.an_gd
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$D_{4}$ \( 1 - 6 T - 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abm
73$D_{4}$ \( 1 - 11 T + 171 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_gp
79$D_{4}$ \( 1 - T - 99 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_adv
83$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_bl
89$D_{4}$ \( 1 - 3 T + 175 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_gt
97$D_{4}$ \( 1 - 17 T + 261 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.97.ar_kb
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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

−10.23930383153632730295190738526, −9.443059319231059430533697548768, −9.179444312415244742936704702838, −8.827338098419392035135335069483, −8.211486246394137926458422211765, −8.157013255232216628929813430220, −7.63909396369849420762098034312, −7.40985582819018280811760839340, −6.78923680454824226029313914357, −6.34395205507217324453852069408, −5.49230567504320095418253672594, −5.36020943825369229757397206023, −5.00369103297796915587654922282, −4.42142255016391846282589323249, −3.86714484341681699273760311749, −3.34898143077619068926443810317, −2.71115258607888438874401649931, −2.24594037488145533393224653218, −1.42776846152052937719727263869, −0.975825374660686359819033943088, 0.975825374660686359819033943088, 1.42776846152052937719727263869, 2.24594037488145533393224653218, 2.71115258607888438874401649931, 3.34898143077619068926443810317, 3.86714484341681699273760311749, 4.42142255016391846282589323249, 5.00369103297796915587654922282, 5.36020943825369229757397206023, 5.49230567504320095418253672594, 6.34395205507217324453852069408, 6.78923680454824226029313914357, 7.40985582819018280811760839340, 7.63909396369849420762098034312, 8.157013255232216628929813430220, 8.211486246394137926458422211765, 8.827338098419392035135335069483, 9.179444312415244742936704702838, 9.443059319231059430533697548768, 10.23930383153632730295190738526

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