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Properties

Label 4-640e2-1.1-c1e2-0-12
Degree $4$
Conductor $409600$
Sign $1$
Analytic cond. $26.1164$
Root an. cond. $2.26062$
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  + 2·5-s − 6·9-s + 12·13-s − 25-s + 4·37-s − 20·41-s − 12·45-s + 14·49-s + 28·53-s + 24·65-s + 27·81-s − 20·89-s − 72·117-s + 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 3.32·13-s − 1/5·25-s + 0.657·37-s − 3.12·41-s − 1.78·45-s + 2·49-s + 3.84·53-s + 2.97·65-s + 3·81-s − 2.11·89-s − 6.65·117-s + 2·121-s − 1.07·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(409600\)    =    \(2^{14} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(26.1164\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 409600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174124652\)
\(L(\frac12)\) \(\approx\) \(2.174124652\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−10.75303934526943902420996081597, −10.35795920313683108427023191836, −10.12808288868102235352532880412, −9.337043635161965010128317803859, −8.760262332511950030321180966448, −8.750650823645753158718143540585, −8.395248025325154176219627201522, −8.014066743832214300646375314543, −7.07199505435686435711251716096, −6.70345157950584420237967460650, −6.08817597828249896892998079615, −5.73681472321274272716718651273, −5.68699361426989069092165727334, −5.05080280077562764934099287219, −3.93973282623656955346345631119, −3.76762221979729374103780599959, −3.10390762895250756848351725549, −2.46820928880916121490131803546, −1.70994289517454519158336061999, −0.846342616596343495164064277755, 0.846342616596343495164064277755, 1.70994289517454519158336061999, 2.46820928880916121490131803546, 3.10390762895250756848351725549, 3.76762221979729374103780599959, 3.93973282623656955346345631119, 5.05080280077562764934099287219, 5.68699361426989069092165727334, 5.73681472321274272716718651273, 6.08817597828249896892998079615, 6.70345157950584420237967460650, 7.07199505435686435711251716096, 8.014066743832214300646375314543, 8.395248025325154176219627201522, 8.750650823645753158718143540585, 8.760262332511950030321180966448, 9.337043635161965010128317803859, 10.12808288868102235352532880412, 10.35795920313683108427023191836, 10.75303934526943902420996081597

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