Properties

Label 4-338e2-1.1-c7e2-0-3
Degree $4$
Conductor $114244$
Sign $1$
Analytic cond. $11148.4$
Root an. cond. $10.2755$
Motivic weight $7$
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  − 78·3-s − 64·4-s + 189·9-s + 4.99e3·12-s + 4.09e3·16-s + 4.20e4·17-s + 1.26e5·23-s + 8.02e3·25-s + 2.74e5·27-s + 2.44e5·29-s − 1.20e4·36-s + 4.04e5·43-s − 3.19e5·48-s + 1.56e6·49-s − 3.27e6·51-s + 3.36e6·53-s − 2.16e6·61-s − 2.62e5·64-s − 2.68e6·68-s − 9.83e6·69-s − 6.25e5·75-s − 1.32e7·79-s − 1.40e7·81-s − 1.90e7·87-s − 8.07e6·92-s − 5.13e5·100-s + 1.08e7·101-s + ⋯
L(s)  = 1  − 1.66·3-s − 1/2·4-s + 7/81·9-s + 0.833·12-s + 1/4·16-s + 2.07·17-s + 2.16·23-s + 0.102·25-s + 2.68·27-s + 1.86·29-s − 0.0432·36-s + 0.774·43-s − 0.416·48-s + 1.89·49-s − 3.45·51-s + 3.10·53-s − 1.22·61-s − 1/8·64-s − 1.03·68-s − 3.60·69-s − 0.171·75-s − 3.01·79-s − 2.92·81-s − 3.10·87-s − 1.08·92-s − 0.0513·100-s + 1.04·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(114244\)    =    \(2^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(11148.4\)
Root analytic conductor: \(10.2755\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 114244,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.065472542\)
\(L(\frac12)\) \(\approx\) \(2.065472542\)
\(L(\frac{9}{2})\) 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)$
bad2$C_2$ \( 1 + p^{6} T^{2} \)
13 \( 1 \)
good3$C_2$ \( ( 1 + 13 p T + p^{7} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 321 p^{2} T^{2} + p^{14} T^{4} \)
7$C_2^2$ \( 1 - 1561237 T^{2} + p^{14} T^{4} \)
11$C_2^2$ \( 1 - 9792738 T^{2} + p^{14} T^{4} \)
17$C_2$ \( ( 1 - 21011 T + p^{7} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 1041033202 T^{2} + p^{14} T^{4} \)
23$C_2$ \( ( 1 - 63072 T + p^{7} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 122238 T + p^{7} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 11596335406 T^{2} + p^{14} T^{4} \)
37$C_2^2$ \( 1 + 5835425375 T^{2} + p^{14} T^{4} \)
41$C_2^2$ \( 1 - 386144547762 T^{2} + p^{14} T^{4} \)
43$C_2$ \( ( 1 - 202025 T + p^{7} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 666901043805 T^{2} + p^{14} T^{4} \)
53$C_2$ \( ( 1 - 1684336 T + p^{7} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 4781381652738 T^{2} + p^{14} T^{4} \)
61$C_2$ \( ( 1 + 1083608 T + p^{7} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 263827378450 T^{2} + p^{14} T^{4} \)
71$C_2^2$ \( 1 - 13844245379757 T^{2} + p^{14} T^{4} \)
73$C_2^2$ \( 1 + 13163740613906 T^{2} + p^{14} T^{4} \)
79$C_2$ \( ( 1 + 6609256 T + p^{7} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 54251727271654 T^{2} + p^{14} T^{4} \)
89$C_2^2$ \( 1 - 39668449289262 T^{2} + p^{14} T^{4} \)
97$C_2^2$ \( 1 - 161556263575582 T^{2} + p^{14} T^{4} \)
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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

−10.48747858526193700949457247822, −10.20109165540770234411833654823, −10.03094107538308766838197331089, −8.890014344999301908129098241912, −8.835422623981705546196043442798, −8.467802719725557571343247557711, −7.44712271987864896372210228205, −7.32796666816401280949529249477, −6.63456018076680221620036822272, −5.98634324063312431265013593240, −5.57056685770835917285746988240, −5.50139508253088040762192221584, −4.62412521713152335115867145626, −4.55108695281147407131610871493, −3.33068289906796800762471707999, −3.10153718910863019290318908148, −2.38652073080089570093399156078, −1.08181423423526317487075699708, −0.833569309101614455452504570622, −0.56367604085787851242333206516, 0.56367604085787851242333206516, 0.833569309101614455452504570622, 1.08181423423526317487075699708, 2.38652073080089570093399156078, 3.10153718910863019290318908148, 3.33068289906796800762471707999, 4.55108695281147407131610871493, 4.62412521713152335115867145626, 5.50139508253088040762192221584, 5.57056685770835917285746988240, 5.98634324063312431265013593240, 6.63456018076680221620036822272, 7.32796666816401280949529249477, 7.44712271987864896372210228205, 8.467802719725557571343247557711, 8.835422623981705546196043442798, 8.890014344999301908129098241912, 10.03094107538308766838197331089, 10.20109165540770234411833654823, 10.48747858526193700949457247822

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