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Properties

Label 4-200e2-1.1-c1e2-0-15
Degree $4$
Conductor $40000$
Sign $1$
Analytic cond. $2.55043$
Root an. cond. $1.26372$
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  + 2·2-s + 2·4-s + 4·7-s + 4·11-s − 8·13-s + 8·14-s − 4·16-s + 8·22-s + 12·23-s − 16·26-s + 8·28-s − 8·32-s − 16·37-s + 4·41-s + 8·44-s + 24·46-s + 4·47-s + 8·49-s − 16·52-s − 8·53-s − 8·64-s − 32·74-s + 16·77-s − 9·81-s + 8·82-s − 32·91-s + 24·92-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.51·7-s + 1.20·11-s − 2.21·13-s + 2.13·14-s − 16-s + 1.70·22-s + 2.50·23-s − 3.13·26-s + 1.51·28-s − 1.41·32-s − 2.63·37-s + 0.624·41-s + 1.20·44-s + 3.53·46-s + 0.583·47-s + 8/7·49-s − 2.21·52-s − 1.09·53-s − 64-s − 3.71·74-s + 1.82·77-s − 81-s + 0.883·82-s − 3.35·91-s + 2.50·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40000\)    =    \(2^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.55043\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.945970608\)
\(L(\frac12)\) \(\approx\) \(2.945970608\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_i
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bg
17$C_2^2$ \( 1 + p^{2} T^{4} \) 2.17.a_a
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.23.am_cu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.37.q_ey
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_i
53$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_bg
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.59.a_da
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2^2$ \( 1 + p^{2} T^{4} \) 2.73.a_a
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.89.a_s
97$C_2^2$ \( 1 + p^{2} T^{4} \) 2.97.a_a
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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

−12.51050852512989233350777280494, −12.26343214044480564145329677017, −12.01058425573912472670383421813, −11.28395701453297447929673370677, −11.12039639515583195070483304497, −10.49501904596233125654467997024, −9.723013269481225824242636840883, −9.157962624803406978300336661028, −8.837796199288290520604214729907, −8.129716278031306848507660235810, −7.23781402617693266289741796492, −7.12512425759943109273312593627, −6.54540621454981498802513765360, −5.48631819650078502909857149801, −5.18572397500845188810684699799, −4.70715514918353254370542366371, −4.25156380407754398270779325137, −3.31387394047788956235381356104, −2.58408328013836953007760711775, −1.61606390462301028453704667568, 1.61606390462301028453704667568, 2.58408328013836953007760711775, 3.31387394047788956235381356104, 4.25156380407754398270779325137, 4.70715514918353254370542366371, 5.18572397500845188810684699799, 5.48631819650078502909857149801, 6.54540621454981498802513765360, 7.12512425759943109273312593627, 7.23781402617693266289741796492, 8.129716278031306848507660235810, 8.837796199288290520604214729907, 9.157962624803406978300336661028, 9.723013269481225824242636840883, 10.49501904596233125654467997024, 11.12039639515583195070483304497, 11.28395701453297447929673370677, 12.01058425573912472670383421813, 12.26343214044480564145329677017, 12.51050852512989233350777280494

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