Properties

Label 4-4400e2-1.1-c1e2-0-25
Degree $4$
Conductor $19360000$
Sign $1$
Analytic cond. $1234.41$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19360000\)    =    \(2^{8} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1234.41\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_b
7$D_{4}$ \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.7.f_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.e_v
23$D_{4}$ \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_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.ai_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.u_he
47$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_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.am_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.i_fu
71$D_{4}$ \( 1 + 6 T - 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_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.b_adv
83$D_{4}$ \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_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

−8.187366243876962278631842569178, −8.116667936934512123062690496697, −7.19864335444048231302553322258, −6.82590778520394867252596398988, −6.67995616845854747281553753748, −6.44548923331337231817826851587, −6.01496045689777599273323556455, −5.89396895749864864133880477882, −5.07420429000274987707134614306, −4.91632919529087544908306014287, −4.55894421730045353922337284346, −3.69930839427493440531947344301, −3.57513545388900530181467248299, −3.36831715496259971684205532511, −2.76852539533625083743296053621, −2.25175248579458075361051496095, −1.52142825204025655523457578620, −1.10968542505886690747660867104, 0, 0, 1.10968542505886690747660867104, 1.52142825204025655523457578620, 2.25175248579458075361051496095, 2.76852539533625083743296053621, 3.36831715496259971684205532511, 3.57513545388900530181467248299, 3.69930839427493440531947344301, 4.55894421730045353922337284346, 4.91632919529087544908306014287, 5.07420429000274987707134614306, 5.89396895749864864133880477882, 6.01496045689777599273323556455, 6.44548923331337231817826851587, 6.67995616845854747281553753748, 6.82590778520394867252596398988, 7.19864335444048231302553322258, 8.116667936934512123062690496697, 8.187366243876962278631842569178

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