Properties

Label 4-338e2-1.1-c7e2-0-4
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 174·3-s − 64·4-s + 1.83e4·9-s + 1.11e4·12-s + 4.09e3·16-s − 1.81e4·17-s − 3.80e4·23-s + 5.32e4·25-s − 1.49e6·27-s + 3.49e5·29-s − 1.17e6·36-s + 6.28e5·43-s − 7.12e5·48-s + 1.61e6·49-s + 3.15e6·51-s − 2.93e6·53-s − 4.79e6·61-s − 2.62e5·64-s + 1.16e6·68-s + 6.61e6·69-s − 9.25e6·75-s + 1.50e6·79-s + 1.02e8·81-s − 6.08e7·87-s + 2.43e6·92-s − 3.40e6·100-s + 3.07e7·101-s + ⋯
L(s)  = 1  − 3.72·3-s − 1/2·4-s + 8.38·9-s + 1.86·12-s + 1/4·16-s − 0.895·17-s − 0.651·23-s + 0.681·25-s − 14.5·27-s + 2.66·29-s − 4.19·36-s + 1.20·43-s − 0.930·48-s + 1.96·49-s + 3.33·51-s − 2.71·53-s − 2.70·61-s − 1/8·64-s + 0.447·68-s + 2.42·69-s − 2.53·75-s + 0.343·79-s + 21.3·81-s − 9.90·87-s + 0.325·92-s − 0.340·100-s + 2.96·101-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: \(2\)
Selberg data: \((4,\ 114244,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_2$ \( ( 1 + 29 p T + p^{7} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 53209 T^{2} + p^{14} T^{4} \)
7$C_2^2$ \( 1 - 1614325 T^{2} + p^{14} T^{4} \)
11$C_2^2$ \( 1 + 21585182 T^{2} + p^{14} T^{4} \)
17$C_2$ \( ( 1 + 9069 T + p^{7} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 407620978 T^{2} + p^{14} T^{4} \)
23$C_2$ \( ( 1 + 19008 T + p^{7} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 174750 T + p^{7} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 54183532078 T^{2} + p^{14} T^{4} \)
37$C_2^2$ \( 1 - 85102132705 T^{2} + p^{14} T^{4} \)
41$C_2^2$ \( 1 + 243012629582 T^{2} + p^{14} T^{4} \)
43$C_2$ \( ( 1 - 314137 T + p^{7} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 813042792445 T^{2} + p^{14} T^{4} \)
53$C_2$ \( ( 1 + 1469232 T + p^{7} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 2327667796738 T^{2} + p^{14} T^{4} \)
61$C_2$ \( ( 1 + 2399608 T + p^{7} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 12117318758290 T^{2} + p^{14} T^{4} \)
71$C_2^2$ \( 1 - 18086309518093 T^{2} + p^{14} T^{4} \)
73$C_2^2$ \( 1 - 2249714370670 T^{2} + p^{14} T^{4} \)
79$C_2$ \( ( 1 - 753560 T + p^{7} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 52785916674790 T^{2} + p^{14} T^{4} \)
89$C_2^2$ \( 1 - 76968332282158 T^{2} + p^{14} T^{4} \)
97$C_2^2$ \( 1 - 158943664213150 T^{2} + p^{14} T^{4} \)
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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

−10.37935337488413392233323810722, −10.14085417719354482936905972893, −9.161308939498770533435175978028, −9.132640414681995362321294894913, −7.79332878846844165681004987439, −7.74792235935024708327052934012, −6.64199861468449579946374845324, −6.60686448897603331567363082071, −6.35468288978839135471083215696, −5.52702163872793654536631412678, −5.43546010140420970936348631065, −4.68157249562387811649174062039, −4.40582965660904932100389888056, −4.19416401318390082760839990914, −2.95943227866243295068341711485, −1.85508830002195248241390960404, −1.05956049128530104561835225235, −0.915897435920530964969315821238, 0, 0, 0.915897435920530964969315821238, 1.05956049128530104561835225235, 1.85508830002195248241390960404, 2.95943227866243295068341711485, 4.19416401318390082760839990914, 4.40582965660904932100389888056, 4.68157249562387811649174062039, 5.43546010140420970936348631065, 5.52702163872793654536631412678, 6.35468288978839135471083215696, 6.60686448897603331567363082071, 6.64199861468449579946374845324, 7.74792235935024708327052934012, 7.79332878846844165681004987439, 9.132640414681995362321294894913, 9.161308939498770533435175978028, 10.14085417719354482936905972893, 10.37935337488413392233323810722

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