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Properties

Label 4-244000-1.1-c1e2-0-9
Degree $4$
Conductor $244000$
Sign $-1$
Analytic cond. $15.5576$
Root an. cond. $1.98603$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 4·9-s − 10-s + 13-s + 16-s + 2·17-s − 4·18-s − 20-s + 25-s + 26-s − 10·29-s + 32-s + 2·34-s − 4·36-s − 22·37-s − 40-s + 4·45-s − 3·49-s + 50-s + 52-s + 6·53-s − 10·58-s + 7·61-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 4/3·9-s − 0.316·10-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.942·18-s − 0.223·20-s + 1/5·25-s + 0.196·26-s − 1.85·29-s + 0.176·32-s + 0.342·34-s − 2/3·36-s − 3.61·37-s − 0.158·40-s + 0.596·45-s − 3/7·49-s + 0.141·50-s + 0.138·52-s + 0.824·53-s − 1.31·58-s + 0.896·61-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 244000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(244000\)    =    \(2^{5} \cdot 5^{3} \cdot 61\)
Sign: $-1$
Analytic conductor: \(15.5576\)
Root analytic conductor: \(1.98603\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 244000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
61$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 8 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.7.a_d
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ab_u
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ac_t
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2^2$ \( 1 - 39 T^{2} + p^{2} T^{4} \) 2.23.a_abn
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.k_cw
31$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.31.a_ay
37$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.w_hm
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.43.a_h
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.53.ag_eh
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
67$C_2^2$ \( 1 - 111 T^{2} + p^{2} T^{4} \) 2.67.a_aeh
71$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \) 2.71.a_ey
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.k_fq
79$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \) 2.79.a_dk
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.83.a_adi
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.89.h_hg
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.97.an_iq
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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

−8.628929762569926660205880139769, −8.408571490757835629058146636142, −7.63624073080312911317235252963, −7.29331265845580693960445095441, −6.85528868560152523617707751234, −6.15194047286183944680216478601, −5.75914806321049392930139336674, −5.20337118481117418379817562909, −4.99674975806443611438491912803, −3.94795329240883412015401848982, −3.61701586749770911627840050699, −3.15369165766996930038763405391, −2.36082179322491379682894367605, −1.56436662930554649630926846640, 0, 1.56436662930554649630926846640, 2.36082179322491379682894367605, 3.15369165766996930038763405391, 3.61701586749770911627840050699, 3.94795329240883412015401848982, 4.99674975806443611438491912803, 5.20337118481117418379817562909, 5.75914806321049392930139336674, 6.15194047286183944680216478601, 6.85528868560152523617707751234, 7.29331265845580693960445095441, 7.63624073080312911317235252963, 8.408571490757835629058146636142, 8.628929762569926660205880139769

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