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Properties

Label 4-384e2-1.1-c1e2-0-0
Degree $4$
Conductor $147456$
Sign $1$
Analytic cond. $9.40192$
Root an. cond. $1.75107$
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  − 8·7-s − 9-s − 4·17-s − 16·23-s + 10·25-s + 8·31-s − 12·41-s + 16·47-s + 34·49-s + 8·63-s + 16·71-s + 12·73-s + 8·79-s + 81-s + 12·89-s − 4·97-s + 8·103-s − 28·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯
L(s)  = 1  − 3.02·7-s − 1/3·9-s − 0.970·17-s − 3.33·23-s + 2·25-s + 1.43·31-s − 1.87·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s + 1.89·71-s + 1.40·73-s + 0.900·79-s + 1/9·81-s + 1.27·89-s − 0.406·97-s + 0.788·103-s − 2.63·113-s + 2.93·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147456\)    =    \(2^{14} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(9.40192\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 147456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5900543520\)
\(L(\frac12)\) \(\approx\) \(0.5900543520\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.53.a_abq
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−11.87155743444823576618691414129, −10.91923308779591399917835911238, −10.47480634983299212945983286650, −10.18122931162721848567067732934, −9.795043619028065311157014176521, −9.289120191022971870579262648242, −9.031461175966213677931290791137, −8.303111151110790689211383562790, −8.022973344092631875977933620594, −7.04387989340392764368635757257, −6.74300109421012252853259468529, −6.30498517475523438238710455379, −6.12222230875314374987170042074, −5.39213430206201839831945637149, −4.58660694737531538003861714253, −3.75572216193165549442230518239, −3.59263299891930224026818578911, −2.70932525750295076442157014770, −2.30795357271118856102465366246, −0.48679498189518907930183266173, 0.48679498189518907930183266173, 2.30795357271118856102465366246, 2.70932525750295076442157014770, 3.59263299891930224026818578911, 3.75572216193165549442230518239, 4.58660694737531538003861714253, 5.39213430206201839831945637149, 6.12222230875314374987170042074, 6.30498517475523438238710455379, 6.74300109421012252853259468529, 7.04387989340392764368635757257, 8.022973344092631875977933620594, 8.303111151110790689211383562790, 9.031461175966213677931290791137, 9.289120191022971870579262648242, 9.795043619028065311157014176521, 10.18122931162721848567067732934, 10.47480634983299212945983286650, 10.91923308779591399917835911238, 11.87155743444823576618691414129

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