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Properties

Label 4-2128e2-1.1-c1e2-0-2
Degree $4$
Conductor $4528384$
Sign $1$
Analytic cond. $288.733$
Root an. cond. $4.12215$
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  + 3·3-s − 6·5-s − 2·7-s + 4·9-s + 5·11-s − 4·13-s − 18·15-s − 7·17-s − 2·19-s − 6·21-s + 6·23-s + 17·25-s + 6·27-s + 9·29-s + 31-s + 15·33-s + 12·35-s − 12·39-s + 5·41-s + 20·43-s − 24·45-s + 2·47-s + 3·49-s − 21·51-s − 3·53-s − 30·55-s − 6·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 2.68·5-s − 0.755·7-s + 4/3·9-s + 1.50·11-s − 1.10·13-s − 4.64·15-s − 1.69·17-s − 0.458·19-s − 1.30·21-s + 1.25·23-s + 17/5·25-s + 1.15·27-s + 1.67·29-s + 0.179·31-s + 2.61·33-s + 2.02·35-s − 1.92·39-s + 0.780·41-s + 3.04·43-s − 3.57·45-s + 0.291·47-s + 3/7·49-s − 2.94·51-s − 0.412·53-s − 4.04·55-s − 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4528384\)    =    \(2^{8} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(288.733\)
Root analytic conductor: \(4.12215\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4528384,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.767892896\)
\(L(\frac12)\) \(\approx\) \(1.767892896\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.3.ad_f
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$D_{4}$ \( 1 - 5 T + 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_z
13$D_{4}$ \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_r
17$D_{4}$ \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_br
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$D_{4}$ \( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_bx
31$D_{4}$ \( 1 - T + 59 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_ch
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.37.a_cj
41$D_{4}$ \( 1 - 5 T + 85 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.41.af_dh
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$D_{4}$ \( 1 - 2 T + 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_br
53$D_{4}$ \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_db
59$D_{4}$ \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_cp
61$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_db
67$D_{4}$ \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_en
71$D_{4}$ \( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_el
73$D_{4}$ \( 1 + 15 T + 121 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.73.p_er
79$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_es
83$D_{4}$ \( 1 - 15 T + 193 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.83.ap_hl
89$D_{4}$ \( 1 + 14 T + 214 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ig
97$D_{4}$ \( 1 - 12 T + 217 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_ij
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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

−8.948756209383605109634339802010, −8.821635425386084083499481360521, −8.576998100754058763523674692105, −8.340133643344082538301705693562, −7.51225691220439603161741445514, −7.49968203350902236162168157837, −7.22275788613926267219417996487, −6.85339261656809563553101462154, −6.23087666245792337841675139062, −6.03830756518747768235414161385, −4.82210746950053998768951198540, −4.47920545596665986957975278263, −4.37733556261980518585125553592, −3.96725132978524647916360178561, −3.39117833565790232960046750912, −3.18334415725561606049348262470, −2.46347910014738945302711450530, −2.45551029367568891214370980542, −1.14539692070843187193209485835, −0.47210294374268222490726203201, 0.47210294374268222490726203201, 1.14539692070843187193209485835, 2.45551029367568891214370980542, 2.46347910014738945302711450530, 3.18334415725561606049348262470, 3.39117833565790232960046750912, 3.96725132978524647916360178561, 4.37733556261980518585125553592, 4.47920545596665986957975278263, 4.82210746950053998768951198540, 6.03830756518747768235414161385, 6.23087666245792337841675139062, 6.85339261656809563553101462154, 7.22275788613926267219417996487, 7.49968203350902236162168157837, 7.51225691220439603161741445514, 8.340133643344082538301705693562, 8.576998100754058763523674692105, 8.821635425386084083499481360521, 8.948756209383605109634339802010

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