Properties

Label 4-2e14-1.1-c1e2-0-2
Degree $4$
Conductor $16384$
Sign $1$
Analytic cond. $1.04465$
Root an. cond. $1.01098$
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  − 2·9-s + 12·17-s + 10·25-s − 12·41-s − 14·49-s − 4·73-s − 5·81-s − 36·89-s − 20·97-s + 36·113-s + 14·121-s + ⋯
L(s)  = 1  − 2/3·9-s + 2.91·17-s + 2·25-s − 1.87·41-s − 2·49-s − 0.468·73-s − 5/9·81-s − 3.81·89-s − 2.03·97-s + 3.38·113-s + 1.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16384\)    =    \(2^{14}\)
Sign: $1$
Analytic conductor: \(1.04465\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16384,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.121646976\)
\(L(\frac12)\) \(\approx\) \(1.121646976\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.37.a_acw
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.83.a_agc
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−13.82686932364188423095463472958, −12.92089563107896110519093354136, −12.44943601301639991206556802577, −12.29818754311778415065996876260, −11.36515988012176547866224364384, −11.29975411628232970704714828288, −10.35406548918393499584234425732, −10.00565361958677993380557336950, −9.588916288933705161532427497691, −8.664242019433138992834480113785, −8.380648284949726943323366842492, −7.76869240934951829307603994212, −7.08971988569328979813016041294, −6.51950332514357795625674257850, −5.58550133389113302414045527469, −5.35336554189979674456025177029, −4.48789758656622716181904592878, −3.23595485619640913873366119897, −3.08634036933586957896715177110, −1.38179389167272920808910458982, 1.38179389167272920808910458982, 3.08634036933586957896715177110, 3.23595485619640913873366119897, 4.48789758656622716181904592878, 5.35336554189979674456025177029, 5.58550133389113302414045527469, 6.51950332514357795625674257850, 7.08971988569328979813016041294, 7.76869240934951829307603994212, 8.380648284949726943323366842492, 8.664242019433138992834480113785, 9.588916288933705161532427497691, 10.00565361958677993380557336950, 10.35406548918393499584234425732, 11.29975411628232970704714828288, 11.36515988012176547866224364384, 12.29818754311778415065996876260, 12.44943601301639991206556802577, 12.92089563107896110519093354136, 13.82686932364188423095463472958

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